theory and practice of algorithmsthomas/tpa/slides/tap-lect04.pdf · having p many elements (up to...
Post on 18-Aug-2020
1 Views
Preview:
TRANSCRIPT
Field Integers Algebraic Structure of Finite Fields End
Theory and Practice of Algorithms
Thomas Zeugmann
Hokkaido UniversityLaboratory for Algorithmics
https://www-alg.ist.hokudai.ac.jp/∼thomas/TPA/
Lecture 4: More About Finite Fields
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IIt remains to find out for which q there are finite Abelianfields Fq. So far, we only know that there are finite Abelianfields Zp, where p is a prime. Clearly, we also interesting inlearning whether or not Zp is the only finite Abelian fieldhaving p many elements (up to isomorphism).Let us consider the additive properties of the element 1; i.e., theidentity element with respect to multiplication. In order toproceed we need the following definition:
Definition 4.1
Let F = (F, +, · ) be any Abelian field. We call the minimum
number c ∈N+ for whichc∑
i=1
1 = 0 the characteristic of F
provided it exists. If there is no c ∈N+ such thatc∑
i=1
1 = 0 then
we define the characteristic of F to be 0.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IIt remains to find out for which q there are finite Abelianfields Fq. So far, we only know that there are finite Abelianfields Zp, where p is a prime. Clearly, we also interesting inlearning whether or not Zp is the only finite Abelian fieldhaving p many elements (up to isomorphism).Let us consider the additive properties of the element 1; i.e., theidentity element with respect to multiplication. In order toproceed we need the following definition:
Definition 4.1
Let F = (F, +, · ) be any Abelian field. We call the minimum
number c ∈N+ for whichc∑
i=1
1 = 0 the characteristic of F
provided it exists. If there is no c ∈N+ such thatc∑
i=1
1 = 0 then
we define the characteristic of F to be 0.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IIRemarks.(1) We denote the characteristic of F by char(F).(2) Note that 1 ∈ F. Since F is field, we also know that
1 + 1 ∈ F, 1 + 1 + 1 ∈ F, and in general thatn∑
i=1
1 ∈ F for
every n ∈N+. So, if F is finite there exists m, n ∈N+,
where m > n such thatm∑
i=1
1 =n∑
i=1
1. Consequently,
m−n∑i=1
1 = 0; i.e., for finite Abelian fields we know that
char(F) is finite.
(3) If F is infinite then char(F) may be zero, e.g., for the realnumbers we have char(R) = 0.
(4) The element 0 and every element of F which can be written
asn∑
i=1
1 for some n ∈N+ is said to be a field integer.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IIRemarks.(1) We denote the characteristic of F by char(F).(2) Note that 1 ∈ F. Since F is field, we also know that
1 + 1 ∈ F, 1 + 1 + 1 ∈ F, and in general thatn∑
i=1
1 ∈ F for
every n ∈N+. So, if F is finite there exists m, n ∈N+,
where m > n such thatm∑
i=1
1 =n∑
i=1
1. Consequently,
m−n∑i=1
1 = 0; i.e., for finite Abelian fields we know that
char(F) is finite.(3) If F is infinite then char(F) may be zero, e.g., for the real
numbers we have char(R) = 0.(4) The element 0 and every element of F which can be written
asn∑
i=1
1 for some n ∈N+ is said to be a field integer.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers III
Theorem 4.1
Let Fq be any finite Abelian field. Then char(Fq) must be a primenumber. The characteristic of an infinite Abelian field is either a primenumber or it is zero. Moreover, if the characteristic of an Abelianfield F is zero then F must be infinite.
Proof. Consider any Abelian field F such that char(F) is finite.Let c = char(F) and suppose that there are k, ` ∈N+ \ {1} such
that c = k`. So k, ` < c and we havek∑
i=1
1 , 0 and∑̀i=1
1 , 0. On
the other hand, it is easy to see that( k∑i=1
1)(∑̀
i=1
1)
=
k∑̀i=1
1 =
c∑i=1
1 = 0 . (1)
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers III
Theorem 4.1
Let Fq be any finite Abelian field. Then char(Fq) must be a primenumber. The characteristic of an infinite Abelian field is either a primenumber or it is zero. Moreover, if the characteristic of an Abelianfield F is zero then F must be infinite.
Proof. Consider any Abelian field F such that char(F) is finite.Let c = char(F) and suppose that there are k, ` ∈N+ \ {1} such
that c = k`. So k, ` < c and we havek∑
i=1
1 , 0 and∑̀i=1
1 , 0. On
the other hand, it is easy to see that( k∑i=1
1)(∑̀
i=1
1)
=
k∑̀i=1
1 =
c∑i=1
1 = 0 . (1)
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IV
Since F is an Abelian field, we know that ab = 0 iff a = 0
or b = 0 (cf. Theorem 1.4, Assertion (2)). Therefore, we must
havek∑
i=1
1 = 0 or∑̀i=1
1 = 0, a contradiction.
Consequently, if the characteristic is finite then it must be prime.
If char(F) = 0 then F cannot be finite (cf. Remark (2)).
Corollary 4.1
Let Fq be any Abelian field. Then the field integers are closed undermultiplication.
Proof. The corollary is a direct consequence of Equation (1).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IV
Since F is an Abelian field, we know that ab = 0 iff a = 0
or b = 0 (cf. Theorem 1.4, Assertion (2)). Therefore, we must
havek∑
i=1
1 = 0 or∑̀i=1
1 = 0, a contradiction.
Consequently, if the characteristic is finite then it must be prime.
If char(F) = 0 then F cannot be finite (cf. Remark (2)).
Corollary 4.1
Let Fq be any Abelian field. Then the field integers are closed undermultiplication.
Proof. The corollary is a direct consequence of Equation (1).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers IV
Since F is an Abelian field, we know that ab = 0 iff a = 0
or b = 0 (cf. Theorem 1.4, Assertion (2)). Therefore, we must
havek∑
i=1
1 = 0 or∑̀i=1
1 = 0, a contradiction.
Consequently, if the characteristic is finite then it must be prime.
If char(F) = 0 then F cannot be finite (cf. Remark (2)).
Corollary 4.1
Let Fq be any Abelian field. Then the field integers are closed undermultiplication.
Proof. The corollary is a direct consequence of Equation (1).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers V
Theorem 4.2
In every Abelian field F of characteristic p, where p ∈N+, the fieldintegers form a subfield of order p isomorphic to the field Zp.
Proof. We know that p must be prime (cf. Theorem 4.1). Let I be
the set of all field integers. Then we havem∑
i=1
1 +n∑
i=1
1 =m+n∑i=1
1.
Since char(F) = p, we directly see that this is just additionmodulo p. In particular, the additive inverse of 0 < m < p
isp−m∑i=1
1. Hence, (I, +) is an Abelian group.
Claim 1. (I \ {0}, · ) is an Abelian group.
By Corollary 4.1 we know that the field integers are closedunder multiplication.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers V
Theorem 4.2
In every Abelian field F of characteristic p, where p ∈N+, the fieldintegers form a subfield of order p isomorphic to the field Zp.
Proof. We know that p must be prime (cf. Theorem 4.1). Let I be
the set of all field integers. Then we havem∑
i=1
1 +n∑
i=1
1 =m+n∑i=1
1.
Since char(F) = p, we directly see that this is just additionmodulo p. In particular, the additive inverse of 0 < m < p
isp−m∑i=1
1. Hence, (I, +) is an Abelian group.
Claim 1. (I \ {0}, · ) is an Abelian group.
By Corollary 4.1 we know that the field integers are closedunder multiplication.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers V
Theorem 4.2
In every Abelian field F of characteristic p, where p ∈N+, the fieldintegers form a subfield of order p isomorphic to the field Zp.
Proof. We know that p must be prime (cf. Theorem 4.1). Let I be
the set of all field integers. Then we havem∑
i=1
1 +n∑
i=1
1 =m+n∑i=1
1.
Since char(F) = p, we directly see that this is just additionmodulo p. In particular, the additive inverse of 0 < m < p
isp−m∑i=1
1. Hence, (I, +) is an Abelian group.
Claim 1. (I \ {0}, · ) is an Abelian group.
By Corollary 4.1 we know that the field integers are closedunder multiplication.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers VI
Hence, it suffices to show that every field integer m ∈ I \ {0}
possesses an inverse element with respect to multiplication.So let m ∈ I \ {0}, m < p, be any field integer. Since p is prime,we conclude that gcd(m, p) = 1. By Theorem 2.5 we know thatthere are x, y ∈ Z such that 1 = mx + py (in Z).
Consequently, in Fwe thus have
1 =( m∑
i=1
1)( x∑
i=1
1)
+( p∑
i=1
1)( y∑
i=1
1)
=( m∑
i=1
1)( x∑
i=1
1)
,
sincep∑
i=1
1 = 0. Sox∑
i=1
1 is the multiplicative inverse ofm∑
i=1
1.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers VI
Hence, it suffices to show that every field integer m ∈ I \ {0}
possesses an inverse element with respect to multiplication.So let m ∈ I \ {0}, m < p, be any field integer. Since p is prime,we conclude that gcd(m, p) = 1. By Theorem 2.5 we know thatthere are x, y ∈ Z such that 1 = mx + py (in Z).
Consequently, in Fwe thus have
1 =( m∑
i=1
1)( x∑
i=1
1)
+( p∑
i=1
1)( y∑
i=1
1)
=( m∑
i=1
1)( x∑
i=1
1)
,
sincep∑
i=1
1 = 0. Sox∑
i=1
1 is the multiplicative inverse ofm∑
i=1
1.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers VII
Furthermore, taking Theorem 2.8 and its proof into account, wesee that the modular inverse in the field Z∗
p is obtained in thesame way as above. Thus, it directly follows that (I \ {0}, · ) isan Abelian group isomorphic to Z∗
p and Claim 1 is shown.
Finally, since F is a field, it is clear that the distributive laws aresatisfied, too. Putting it all together, the theorem follows.
Next, we show that every Abelian field of characteristic p
inherits a considerable part of its structure from its subfieldformed by its field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers VII
Furthermore, taking Theorem 2.8 and its proof into account, wesee that the modular inverse in the field Z∗
p is obtained in thesame way as above. Thus, it directly follows that (I \ {0}, · ) isan Abelian group isomorphic to Z∗
p and Claim 1 is shown.Finally, since F is a field, it is clear that the distributive laws aresatisfied, too. Putting it all together, the theorem follows.
Next, we show that every Abelian field of characteristic p
inherits a considerable part of its structure from its subfieldformed by its field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Field Integers VII
Furthermore, taking Theorem 2.8 and its proof into account, wesee that the modular inverse in the field Z∗
p is obtained in thesame way as above. Thus, it directly follows that (I \ {0}, · ) isan Abelian group isomorphic to Z∗
p and Claim 1 is shown.Finally, since F is a field, it is clear that the distributive laws aresatisfied, too. Putting it all together, the theorem follows.
Next, we show that every Abelian field of characteristic p
inherits a considerable part of its structure from its subfieldformed by its field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields I
Theorem 4.3
In every Abelian field F of characteristic p, p ∈N+, the equation
(x − a)p = xp − ap
is satisfied for all x, a ∈ F.
Proof. By the binomial theorem we have
(x − a)p =
p∑k=0
(p
k
)xp−k(−a)k .
Taking into account that(p0
)=
(pp
)= 1 and that
(pk
)≡ 0 mod p
for all 0 < k < p (cf. Theorem 4.2), the theorem follows (notethat in Z2 we have −1 = +1).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields I
Theorem 4.3
In every Abelian field F of characteristic p, p ∈N+, the equation
(x − a)p = xp − ap
is satisfied for all x, a ∈ F.
Proof. By the binomial theorem we have
(x − a)p =
p∑k=0
(p
k
)xp−k(−a)k .
Taking into account that(p0
)=
(pp
)= 1 and that
(pk
)≡ 0 mod p
for all 0 < k < p (cf. Theorem 4.2), the theorem follows (notethat in Z2 we have −1 = +1).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields II
Theorem 4.3 directly allows for the following corollaries:
Corollary 4.2
In every finite Abelian field F of characteristic p there does not existany element that has an order pk, where k ∈N+, k > 1.
Proof. Suppose the converse. Then there exists an element a ∈ F
such that apk = 1. Since ord(a) = pk, we know that ak , 1.Consequently, we have ak − 1 , 0. By Theorem 1.4,Assertion (2) we thus have (ak − 1)p , 0. On the other hand, byTheorem 4.3 we obtain (ak − 1)p = apk − 1p = 0, acontradiction.
In the following, we always asume that char(F) , 0.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields II
Theorem 4.3 directly allows for the following corollaries:
Corollary 4.2
In every finite Abelian field F of characteristic p there does not existany element that has an order pk, where k ∈N+, k > 1.
Proof. Suppose the converse. Then there exists an element a ∈ F
such that apk = 1. Since ord(a) = pk, we know that ak , 1.Consequently, we have ak − 1 , 0. By Theorem 1.4,Assertion (2) we thus have (ak − 1)p , 0. On the other hand, byTheorem 4.3 we obtain (ak − 1)p = apk − 1p = 0, acontradiction.
In the following, we always asume that char(F) , 0.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields III
Corollary 4.3
Let w1, . . . , wk be any elements of an Abelian field F of
characteristic p. Then we have( k∑
i=1
wi
)pn
=( k∑
i=1
wpn
i
)for
all n ∈N.
Proof. The proof is by induction. For k = 1 the assertion isobvious. So, let k = 2 and let us perform induction over n. Forthe induction basis, i.e., for n = 0, we have p0 = 1 and theequality is again obvious.
The induction step is from n to n + 1. Using the inductionhypothesis we directly obtain
(w1 + w2)pn+1=
((w1 + w2)pn
)p=
(w
pn
1 + wpn
2
)p.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields III
Corollary 4.3
Let w1, . . . , wk be any elements of an Abelian field F of
characteristic p. Then we have( k∑
i=1
wi
)pn
=( k∑
i=1
wpn
i
)for
all n ∈N.
Proof. The proof is by induction. For k = 1 the assertion isobvious. So, let k = 2 and let us perform induction over n. Forthe induction basis, i.e., for n = 0, we have p0 = 1 and theequality is again obvious.
The induction step is from n to n + 1. Using the inductionhypothesis we directly obtain
(w1 + w2)pn+1=
((w1 + w2)pn
)p=
(w
pn
1 + wpn
2
)p.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields III
Corollary 4.3
Let w1, . . . , wk be any elements of an Abelian field F of
characteristic p. Then we have( k∑
i=1
wi
)pn
=( k∑
i=1
wpn
i
)for
all n ∈N.
Proof. The proof is by induction. For k = 1 the assertion isobvious. So, let k = 2 and let us perform induction over n. Forthe induction basis, i.e., for n = 0, we have p0 = 1 and theequality is again obvious.
The induction step is from n to n + 1. Using the inductionhypothesis we directly obtain
(w1 + w2)pn+1=
((w1 + w2)pn
)p=
(w
pn
1 + wpn
2
)p.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields IV
Next, we apply Theorem 4.3 for x = wpn
1 and a = −wpn
2 .Consequently, we have(
wpn
1 + wpn
2
)p=
(w
pn
1
)p−
(−w
pn
2
)p
= wpn+1
1 + wpn+1
2 ,
and the induction step is shown.
Next, we fix n and perform the induction step from k to k + 1.
Taking into account thatk∑
i=1
wi ∈ F we can apply the already
shown part. Then we use the induction hypothesis for k.Consequently, we obtain the following:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields IV
Next, we apply Theorem 4.3 for x = wpn
1 and a = −wpn
2 .Consequently, we have(
wpn
1 + wpn
2
)p=
(w
pn
1
)p−
(−w
pn
2
)p
= wpn+1
1 + wpn+1
2 ,
and the induction step is shown.
Next, we fix n and perform the induction step from k to k + 1.
Taking into account thatk∑
i=1
wi ∈ F we can apply the already
shown part. Then we use the induction hypothesis for k.Consequently, we obtain the following:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields V
(k+1∑i=1
wi
)pn
=( k∑
i=1
wi + wk+1
)pn
=( k∑
i=1
wi
)pn
+ wpn
k+1
=
k∑i=1
wpn
i + wpn
k+1 =
k+1∑i=1
wpn
i ,
and the corollary is shown.
Furthermore, Corollary 4.3 implies the following special case ofFermat’s theorem:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields V
(k+1∑i=1
wi
)pn
=( k∑
i=1
wi + wk+1
)pn
=( k∑
i=1
wi
)pn
+ wpn
k+1
=
k∑i=1
wpn
i + wpn
k+1 =
k+1∑i=1
wpn
i ,
and the corollary is shown.
Furthermore, Corollary 4.3 implies the following special case ofFermat’s theorem:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VI
Corollary 4.4
Let F be any Abelian field of characteristic p, and let k be any fieldinteger. Then we have kpn
= k for all n ∈N.
Proof. Since k is a field integer, we have k =k∑
i=1
1. Hence, by
Corollary 4.3 we directly obtain that
kpn=
( k∑i=1
1)pn
=
k∑i=1
1pn
=
k∑i=1
1 = k .
Thus, the corollary is shown.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VI
Corollary 4.4
Let F be any Abelian field of characteristic p, and let k be any fieldinteger. Then we have kpn
= k for all n ∈N.
Proof. Since k is a field integer, we have k =k∑
i=1
1. Hence, by
Corollary 4.3 we directly obtain that
kpn=
( k∑i=1
1)pn
=
k∑i=1
1pn
=
k∑i=1
1 = k .
Thus, the corollary is shown.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VII
Now, we are in a position to show the followingcharacterization of field integers:
Theorem 4.4
Let F be any Abelian field of characteristic p. Then for every a ∈ F wehave the following: The element a is a field integer in F iff it is asolution of the equation xp − x = 0.
Proof. Necessity. If a is field integer then by Corollary 4.4 wehave ap = a. Consequently, a is a solution of xp − x = 0.
Sufficiency. The polynomial xp − x = 0 has degree p. Thus, inaccordance with Theorem 3.5 we know that it has at most p
many zeros. Since there are p many field integers and sinceevery field integer is a zero of xp − x = 0, we are done.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VII
Now, we are in a position to show the followingcharacterization of field integers:
Theorem 4.4
Let F be any Abelian field of characteristic p. Then for every a ∈ F wehave the following: The element a is a field integer in F iff it is asolution of the equation xp − x = 0.
Proof. Necessity. If a is field integer then by Corollary 4.4 wehave ap = a. Consequently, a is a solution of xp − x = 0.
Sufficiency. The polynomial xp − x = 0 has degree p. Thus, inaccordance with Theorem 3.5 we know that it has at most p
many zeros. Since there are p many field integers and sinceevery field integer is a zero of xp − x = 0, we are done.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VII
Now, we are in a position to show the followingcharacterization of field integers:
Theorem 4.4
Let F be any Abelian field of characteristic p. Then for every a ∈ F wehave the following: The element a is a field integer in F iff it is asolution of the equation xp − x = 0.
Proof. Necessity. If a is field integer then by Corollary 4.4 wehave ap = a. Consequently, a is a solution of xp − x = 0.
Sufficiency. The polynomial xp − x = 0 has degree p. Thus, inaccordance with Theorem 3.5 we know that it has at most p
many zeros. Since there are p many field integers and sinceevery field integer is a zero of xp − x = 0, we are done.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VIII
Corollary 4.5
Let F be any Abelian field of characteristic p, and let w ∈ F be suchthat w is not a field integer. Then we have wp , w.
Nevertheless, wp is still closely related to w as the followingtheorem shows:
Theorem 4.5
Let F be any Abelian field of characteristic p, let f ∈ Zp[x], andlet w ∈ F be such that f(w) = 0. Then we have f
(wpn)
= 0 forall n ∈N.
Proof. Let deg(f) = d > 0 and f0, . . . , fd ∈ Zp be the coefficientsof f. Then we know that the fi, i = 0, . . . , d, are field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VIII
Corollary 4.5
Let F be any Abelian field of characteristic p, and let w ∈ F be suchthat w is not a field integer. Then we have wp , w.
Nevertheless, wp is still closely related to w as the followingtheorem shows:
Theorem 4.5
Let F be any Abelian field of characteristic p, let f ∈ Zp[x], andlet w ∈ F be such that f(w) = 0. Then we have f
(wpn)
= 0 forall n ∈N.
Proof. Let deg(f) = d > 0 and f0, . . . , fd ∈ Zp be the coefficientsof f. Then we know that the fi, i = 0, . . . , d, are field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields VIII
Corollary 4.5
Let F be any Abelian field of characteristic p, and let w ∈ F be suchthat w is not a field integer. Then we have wp , w.
Nevertheless, wp is still closely related to w as the followingtheorem shows:
Theorem 4.5
Let F be any Abelian field of characteristic p, let f ∈ Zp[x], andlet w ∈ F be such that f(w) = 0. Then we have f
(wpn)
= 0 forall n ∈N.
Proof. Let deg(f) = d > 0 and f0, . . . , fd ∈ Zp be the coefficientsof f. Then we know that the fi, i = 0, . . . , d, are field integers.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields IX
By assumption we have f(w) = 0. Consequently, byCorollary 4.4 we know that f
pn
i = fi for all i = 0, . . . , d.Therefore, Corollary 4.3 directly yields
0 =( d∑
i=0
fiwi)pn
=
d∑i=0
(fiw
i)pn
=
d∑i=0
fpn
i wipn=
d∑i=0
fi
(wpn
)i= f
(wpn
),
i.e., we have f(wpn)
= 0.
Looking at w, wp, wp2, wp3
, . . . we see that they form a subsetof the powers of w. Hence, the number of different elements ofthis form solely depends on the order of w.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields IX
By assumption we have f(w) = 0. Consequently, byCorollary 4.4 we know that f
pn
i = fi for all i = 0, . . . , d.Therefore, Corollary 4.3 directly yields
0 =( d∑
i=0
fiwi)pn
=
d∑i=0
(fiw
i)pn
=
d∑i=0
fpn
i wipn=
d∑i=0
fi
(wpn
)i= f
(wpn
),
i.e., we have f(wpn)
= 0.
Looking at w, wp, wp2, wp3
, . . . we see that they form a subsetof the powers of w. Hence, the number of different elements ofthis form solely depends on the order of w.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields X
Theorem 4.6
Let F be any finite Abelian field F of characteristic p, let w ∈ F be anyelement of order n. Furthermore, let m be the order of p in Z∗
n. Thenwe have wpm
= w, and the m elements w, wp, wp2, . . . , wpm−1
arepairwise distinct.
Proof. By assumption, pm ≡ 1 mod n and ps . 1 mod n forall 0 < s < m. Furthermore, it holds that
wpk= wpi ⇐⇒ wpk−pi
= 1 ⇐⇒ pk − pi = ` · n⇐⇒ pk ≡ pi mod n ⇐⇒ pk−i ≡ 1 mod n ,
i.e., k − i must be a multiple of m.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields X
Theorem 4.6
Let F be any finite Abelian field F of characteristic p, let w ∈ F be anyelement of order n. Furthermore, let m be the order of p in Z∗
n. Thenwe have wpm
= w, and the m elements w, wp, wp2, . . . , wpm−1
arepairwise distinct.
Proof. By assumption, pm ≡ 1 mod n and ps . 1 mod n forall 0 < s < m. Furthermore, it holds that
wpk= wpi ⇐⇒ wpk−pi
= 1 ⇐⇒ pk − pi = ` · n⇐⇒ pk ≡ pi mod n ⇐⇒ pk−i ≡ 1 mod n ,
i.e., k − i must be a multiple of m.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XI
Theorem 4.7
Let w be an element of order n in a finite Abelian field Fq ofcharacteristic p, and let m be the order of p in Z∗
n. Then the
coefficients of the mth degree polynomial f(x) =m−1∏i=0
(x − wpi
)are
field integers. Furthermore, f is irreducible in Zp[x].
Proof. By assumption we have pm ≡ 1 mod n and wn = 1 (inthe field Fq). Thus, we see that wpm
= w = wp0. By
Theorem 4.3 we know that(x − wpi
)p= xp − wpi+1
forall i = 0, . . . ,m − 1. Consequently,
(f(x))p =
m−1∏i=0
(x − wpi
)p=
m−1∏i=0
(xp − wpi+1
).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XI
Theorem 4.7
Let w be an element of order n in a finite Abelian field Fq ofcharacteristic p, and let m be the order of p in Z∗
n. Then the
coefficients of the mth degree polynomial f(x) =m−1∏i=0
(x − wpi
)are
field integers. Furthermore, f is irreducible in Zp[x].
Proof. By assumption we have pm ≡ 1 mod n and wn = 1 (inthe field Fq). Thus, we see that wpm
= w = wp0. By
Theorem 4.3 we know that(x − wpi
)p= xp − wpi+1
forall i = 0, . . . ,m − 1. Consequently,
(f(x))p =
m−1∏i=0
(x − wpi
)p=
m−1∏i=0
(xp − wpi+1
).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XII
Next, we manipulate the index of the product as follows:
m−1∏i=0
(xp − wpi+1
)=
m∏i=1
(xp − wpi
)=
m−1∏i=0
(xp − wpi
),
where in the last step we used xp − wpm= xp − w = xp − wp0
.
Thus, we conclude that (f(x))p = f(xp).
Writing f as f(x) =m∑
i=0
fixi we therefore obtain
(f(x))p =( m∑
i=0
fixi)p
=
m∑i=0
fpi xpi
,
where the last step is by Corollary 4.3.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XII
Next, we manipulate the index of the product as follows:
m−1∏i=0
(xp − wpi+1
)=
m∏i=1
(xp − wpi
)=
m−1∏i=0
(xp − wpi
),
where in the last step we used xp − wpm= xp − w = xp − wp0
.
Thus, we conclude that (f(x))p = f(xp).
Writing f as f(x) =m∑
i=0
fixi we therefore obtain
(f(x))p =( m∑
i=0
fixi)p
=
m∑i=0
fpi xpi
,
where the last step is by Corollary 4.3.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XIII
Furthermore, f(xp) =m∑
i=0
fpi xpi
, and as shown above
(f(x))p = f(xp). Consequently, we see that fpi = fi must hold
(cf. our definition of equality for polynomials). By Theorem 4.4we see that fi must be a field integer for all i = 0, . . . ,m.
It remains to show that f is irreducible in Zp[x]. Note that f ismonic. Suppose the converse. Then there are monicpolynomials g, h ∈ Zp[x] such that f(x) = g(x)h(x).
If g(w) = 0 and if all coefficients of g are field integer then wealso have g (wp) = 0, . . . , g
(wpm−1
)= 0 (cf. Theorem 4.5).
Hence, Theorem 4.6 implies deg(g) = m, and so f = g.Analogously one sees that h(w) = 0 implies f = h.Consequently, the polynomial f is irreducible in Zp[x].
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XIII
Furthermore, f(xp) =m∑
i=0
fpi xpi
, and as shown above
(f(x))p = f(xp). Consequently, we see that fpi = fi must hold
(cf. our definition of equality for polynomials). By Theorem 4.4we see that fi must be a field integer for all i = 0, . . . ,m.
It remains to show that f is irreducible in Zp[x]. Note that f ismonic. Suppose the converse. Then there are monicpolynomials g, h ∈ Zp[x] such that f(x) = g(x)h(x).
If g(w) = 0 and if all coefficients of g are field integer then wealso have g (wp) = 0, . . . , g
(wpm−1
)= 0 (cf. Theorem 4.5).
Hence, Theorem 4.6 implies deg(g) = m, and so f = g.Analogously one sees that h(w) = 0 implies f = h.Consequently, the polynomial f is irreducible in Zp[x].
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XIV
Remarks. Since all polynomials over the field integers forwhich w is a root must also have wp, wp2
, . . . , wpm−1as roots,
we conclude that all such polynomials are multiples of the
polynomial f(x) =m−1∏i=0
(x − wpi
).
We call f the minimal polynomial of w, and its degree m is said tobe the degree of w. Since all the elements wp, wp2
, . . . , wpm−1
must have the same minimal polynomial as w, we call theseelements the conjugates of w.
Compare this to the complex numbers i and −i which are alsosaid to be conjugates. These complex numbers are the rootsof x2 + 1. The set of all complex numbers is then the set of alllinear combinations a + bi, where a, b ∈ R. They are added asvectors, i.e., (a + bi) + (c + di) = (a + c) + (b + d)i, andmultiplied as polynomials in i modulo i2 + 1; i.e., we have
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XIV
Remarks. Since all polynomials over the field integers forwhich w is a root must also have wp, wp2
, . . . , wpm−1as roots,
we conclude that all such polynomials are multiples of the
polynomial f(x) =m−1∏i=0
(x − wpi
).
We call f the minimal polynomial of w, and its degree m is said tobe the degree of w. Since all the elements wp, wp2
, . . . , wpm−1
must have the same minimal polynomial as w, we call theseelements the conjugates of w.
Compare this to the complex numbers i and −i which are alsosaid to be conjugates. These complex numbers are the rootsof x2 + 1. The set of all complex numbers is then the set of alllinear combinations a + bi, where a, b ∈ R. They are added asvectors, i.e., (a + bi) + (c + di) = (a + c) + (b + d)i, andmultiplied as polynomials in i modulo i2 + 1; i.e., we have
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XIV
Remarks. Since all polynomials over the field integers forwhich w is a root must also have wp, wp2
, . . . , wpm−1as roots,
we conclude that all such polynomials are multiples of the
polynomial f(x) =m−1∏i=0
(x − wpi
).
We call f the minimal polynomial of w, and its degree m is said tobe the degree of w. Since all the elements wp, wp2
, . . . , wpm−1
must have the same minimal polynomial as w, we call theseelements the conjugates of w.
Compare this to the complex numbers i and −i which are alsosaid to be conjugates. These complex numbers are the rootsof x2 + 1. The set of all complex numbers is then the set of alllinear combinations a + bi, where a, b ∈ R. They are added asvectors, i.e., (a + bi) + (c + di) = (a + c) + (b + d)i, andmultiplied as polynomials in i modulo i2 + 1; i.e., we have
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XV
(a + bi)(c + di) = (ac − bd) + (ad + bc)i, since i2 : (i2 + 1) hasremainder −1.
Since the minimal polynomial of w is an irreducible polynomialof degree m over the field integers, the pm polynomials in w ofdegree less than m over the field integers are all different. Theyform a field.
Theorem 4.8
If w is a field element of degree m in a finite Abelian field Fq ofcharacteristic p then the polynomials over the field integers of Fq ofdegree less than m in w form a subfield of Fq that has order pm.
Proof. First, the pm polynomials in w of degree less than m overthe field integers are all distinct. This can be seen as follows:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XV
(a + bi)(c + di) = (ac − bd) + (ad + bc)i, since i2 : (i2 + 1) hasremainder −1.
Since the minimal polynomial of w is an irreducible polynomialof degree m over the field integers, the pm polynomials in w ofdegree less than m over the field integers are all different. Theyform a field.
Theorem 4.8
If w is a field element of degree m in a finite Abelian field Fq ofcharacteristic p then the polynomials over the field integers of Fq ofdegree less than m in w form a subfield of Fq that has order pm.
Proof. First, the pm polynomials in w of degree less than m overthe field integers are all distinct. This can be seen as follows:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XV
(a + bi)(c + di) = (ac − bd) + (ad + bc)i, since i2 : (i2 + 1) hasremainder −1.
Since the minimal polynomial of w is an irreducible polynomialof degree m over the field integers, the pm polynomials in w ofdegree less than m over the field integers are all different. Theyform a field.
Theorem 4.8
If w is a field element of degree m in a finite Abelian field Fq ofcharacteristic p then the polynomials over the field integers of Fq ofdegree less than m in w form a subfield of Fq that has order pm.
Proof. First, the pm polynomials in w of degree less than m overthe field integers are all distinct. This can be seen as follows:
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVISuppose that two of these polynomials are equal. Then w
would be a root of their difference. But the difference is apolynomial of degree less than m, a contradiction to theassumption that the degree of w is m.
The sum of two polynomials of degree less than m in w is againa polynomial of degree less than m in w. The product of twopolynomials of degree less than m in w is a polynomial in w
which can be reduced modulo the minimal polynomial of w,i.e., the product is also a polynomial of degree less than m in w.The multiplicative inverse of a polynomial h of degree less thanm in w can be found by using the ECL for h and the minimalpolynomial of w. Since the minimal polynomial f of w isirreducible, we have gcd(h, f) = 1. By Bézout’s lemma thereare u, v such that 1 = h(x)u(x) + f(x)v(x). Consequently, wehave h(x)u(x) ≡ 1 mod f(x), and u(w) is the multiplicativeinverse of h(w).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVISuppose that two of these polynomials are equal. Then w
would be a root of their difference. But the difference is apolynomial of degree less than m, a contradiction to theassumption that the degree of w is m.The sum of two polynomials of degree less than m in w is againa polynomial of degree less than m in w. The product of twopolynomials of degree less than m in w is a polynomial in w
which can be reduced modulo the minimal polynomial of w,i.e., the product is also a polynomial of degree less than m in w.
The multiplicative inverse of a polynomial h of degree less thanm in w can be found by using the ECL for h and the minimalpolynomial of w. Since the minimal polynomial f of w isirreducible, we have gcd(h, f) = 1. By Bézout’s lemma thereare u, v such that 1 = h(x)u(x) + f(x)v(x). Consequently, wehave h(x)u(x) ≡ 1 mod f(x), and u(w) is the multiplicativeinverse of h(w).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVISuppose that two of these polynomials are equal. Then w
would be a root of their difference. But the difference is apolynomial of degree less than m, a contradiction to theassumption that the degree of w is m.The sum of two polynomials of degree less than m in w is againa polynomial of degree less than m in w. The product of twopolynomials of degree less than m in w is a polynomial in w
which can be reduced modulo the minimal polynomial of w,i.e., the product is also a polynomial of degree less than m in w.The multiplicative inverse of a polynomial h of degree less thanm in w can be found by using the ECL for h and the minimalpolynomial of w. Since the minimal polynomial f of w isirreducible, we have gcd(h, f) = 1. By Bézout’s lemma thereare u, v such that 1 = h(x)u(x) + f(x)v(x). Consequently, wehave h(x)u(x) ≡ 1 mod f(x), and u(w) is the multiplicativeinverse of h(w).
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVII
Note that every element a of this subfield can be written
as a =m−1∑i=0
aiwi, where ai ∈ Zp.
We show that every finite Abelian field must be of this type.
Theorem 4.9
The order of every finite Abelian field is a power of its characteristic.
Proof. Let Fq be any finite Abelian field, and let p = char(Fq).By Theorem 3.6 we know that F∗q has a generator g such thatord(g) = q − 1. The number of conjugates of g is equal to themultiplicative order, say m, of p modulo q − 1. Since(q − 1)|(pm − 1), we conclude that q − 1 6 pm − 1, i.e., wehave q 6 pm.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVII
Note that every element a of this subfield can be written
as a =m−1∑i=0
aiwi, where ai ∈ Zp.
We show that every finite Abelian field must be of this type.
Theorem 4.9
The order of every finite Abelian field is a power of its characteristic.
Proof. Let Fq be any finite Abelian field, and let p = char(Fq).By Theorem 3.6 we know that F∗q has a generator g such thatord(g) = q − 1. The number of conjugates of g is equal to themultiplicative order, say m, of p modulo q − 1. Since(q − 1)|(pm − 1), we conclude that q − 1 6 pm − 1, i.e., wehave q 6 pm.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVII
Note that every element a of this subfield can be written
as a =m−1∑i=0
aiwi, where ai ∈ Zp.
We show that every finite Abelian field must be of this type.
Theorem 4.9
The order of every finite Abelian field is a power of its characteristic.
Proof. Let Fq be any finite Abelian field, and let p = char(Fq).By Theorem 3.6 we know that F∗q has a generator g such thatord(g) = q − 1. The number of conjugates of g is equal to themultiplicative order, say m, of p modulo q − 1. Since(q − 1)|(pm − 1), we conclude that q − 1 6 pm − 1, i.e., wehave q 6 pm.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVIII
Finally, the degree of g is equal to m (cf. Theorem 4.7).Consequently, as shown above, we know that the pm
polynomials of degree less than m are all distinct. Therefore,we also have pm 6 q, and thus we conclude q = pm.
Remarks. We have shown that there is a finite Abelian field Fq
if and only if q = pm, where p is a prime and m ∈N+.However, we still have to prove that the finite field Fq isuniquely determined (up to isomorphism).Furthermore, we also obtained a method to construct a finitefield Fq. That is, one starts from Zp and has to find anirreducible polynomial of degree m over Zp. We exemplify themethod below.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Algebraic Structure of Finite Fields XVIII
Finally, the degree of g is equal to m (cf. Theorem 4.7).Consequently, as shown above, we know that the pm
polynomials of degree less than m are all distinct. Therefore,we also have pm 6 q, and thus we conclude q = pm.
Remarks. We have shown that there is a finite Abelian field Fq
if and only if q = pm, where p is a prime and m ∈N+.However, we still have to prove that the finite field Fq isuniquely determined (up to isomorphism).Furthermore, we also obtained a method to construct a finitefield Fq. That is, one starts from Zp and has to find anirreducible polynomial of degree m over Zp. We exemplify themethod below.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Example
Example 4.1.We want to construct a finite field having 9 elements. Since9 = 32, we need a polynomial f of degree 2 which is irreducibleover Z3. For that purpose we can take f(x) =df x2 − x + 2
which is irreducible over Z3, since we have f(0) ≡ 2 mod 3,f(1) ≡ 2 mod 3, and f(2) ≡ 1 mod 3. Now, the elements of F9
can be expressed as aϑ + b, where a, b ∈ Z3, using an elementϑ satisfying ϑ2 − ϑ + 2 = 0.
That is, we obtain the 9 elements:0, 1, 2, ϑ, 2ϑ, ϑ + 1, ϑ + 2, 2ϑ + 1, and 2ϑ + 2. Thecomputation with these elements is performed in the same wayas computations with polynomials mod ϑ2 − ϑ + 2 therebyreducing the coefficients modulo 3.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Example
Example 4.1.We want to construct a finite field having 9 elements. Since9 = 32, we need a polynomial f of degree 2 which is irreducibleover Z3. For that purpose we can take f(x) =df x2 − x + 2
which is irreducible over Z3, since we have f(0) ≡ 2 mod 3,f(1) ≡ 2 mod 3, and f(2) ≡ 1 mod 3. Now, the elements of F9
can be expressed as aϑ + b, where a, b ∈ Z3, using an elementϑ satisfying ϑ2 − ϑ + 2 = 0.
That is, we obtain the 9 elements:0, 1, 2, ϑ, 2ϑ, ϑ + 1, ϑ + 2, 2ϑ + 1, and 2ϑ + 2. Thecomputation with these elements is performed in the same wayas computations with polynomials mod ϑ2 − ϑ + 2 therebyreducing the coefficients modulo 3.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Example cont.
Thus, we obtain the following multiplication table:
· 1 2 ϑ 2ϑ ϑ+1 ϑ+2 2ϑ+1 2ϑ+2
1 1 2 ϑ 2ϑ ϑ+1 ϑ+2 2ϑ+1 2ϑ+2
2 2 1 2ϑ ϑ 2ϑ+2 2ϑ+1 ϑ+2 ϑ+1
ϑ ϑ 2ϑ ϑ+1 2ϑ+2 2ϑ+1 1 2 ϑ+2
2ϑ 2ϑ ϑ 2ϑ+2 ϑ+1 ϑ+2 2 1 2ϑ+1
ϑ+1 ϑ+1 2ϑ+2 2ϑ+1 ϑ+2 2 ϑ 2ϑ 1
ϑ+2 ϑ+2 2ϑ+1 1 2 ϑ 2ϑ+2 ϑ+1 2ϑ
2ϑ+1 2ϑ+1 ϑ+2 2 1 2ϑ ϑ+1 2ϑ+2 ϑ
2ϑ+2 2ϑ+2 ϑ+1 ϑ+2 2ϑ+1 1 2ϑ ϑ 2
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Example cont.
As an example, we provide the computation of the entry in row2ϑ + 2 and column 2ϑ + 1. We multiply the polynomials 2ϑ + 2
and 2ϑ + 1, reduce the result modulo ϑ2 − ϑ + 2, and thecoefficients modulo 3. Thus, we obtain:
(2ϑ + 2)(2ϑ + 1) = 4ϑ2 + 2ϑ + 4ϑ + 2
= ϑ2 + 2 ,
and(ϑ2 + 2) : (ϑ2 − ϑ + 2) = 1
−(ϑ2 − ϑ + 2)
ϑ
Thus, the remainder is ϑ as already displayed in themultiplication table.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Example cont.
As an example, we provide the computation of the entry in row2ϑ + 2 and column 2ϑ + 1. We multiply the polynomials 2ϑ + 2
and 2ϑ + 1, reduce the result modulo ϑ2 − ϑ + 2, and thecoefficients modulo 3. Thus, we obtain:
(2ϑ + 2)(2ϑ + 1) = 4ϑ2 + 2ϑ + 4ϑ + 2
= ϑ2 + 2 ,
and(ϑ2 + 2) : (ϑ2 − ϑ + 2) = 1
−(ϑ2 − ϑ + 2)
ϑ
Thus, the remainder is ϑ as already displayed in themultiplication table.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Exercises
Exercise 1. Find an irreducible polynomial of degree 2 over Z7.
Exercise 2. Find out whether or not there is a finite field having 8
elements. In case your answer is affirmative, construct such a finitefield.
Exercise 3. Find out whether or not there is a finite field having 27,36, 51, and 2401 elements, respectively. Justify your answer.
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Thank you!
Theory and Practice of Algorithms c©Thomas Zeugmann
Field Integers Algebraic Structure of Finite Fields End
Pierre de Fermat
Theory and Practice of Algorithms c©Thomas Zeugmann
top related