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235

INDEX

A

Adele ring, 66Adelization, 174–175Algebra, 47, 79Algebraic integer, 27Algebraic number field, 27Ambiguous ideal, 229Ambiguous ideal class, 229Anisotropic, 115Anti-automorphism, 89Anti-isomorphism, 89Archimedean prime, 66Archimedean valuation, 16

B

Binary Form, 203

C

Canonical automorphism, 122Canonical involution, 122Cauchy sequence, 16Center, 86Central simple algebra, 86ffCentral, 86Characteristic algebra, 153Character (modulo an integer), 7Class (of a binary form), 204

Class (of a lattice), 171Class (of a matrix), 178Class (of a proper ideal), 215Class number (of a number field), 42Class number (of an order), 215Class number (of a quadratic space), 181Clifford algebra, 121Clifford group, 127Commutor, 90Complete, 16Completely reducible (module), 81Completion, 17, 39Conductor (of a character), 9Conductor (of an order), 213Core dimension, 117Core subspace, 117Cyclotomic field, 75ff

D

Decomposition group, 71

Dedekind zeta function, 56Definite (quaternion algebra), 147Different, 59Dirichlet character, 7Discrete order function, 15

Discriminant algebra, 120Discriminant field, 120Discriminant ideal (of a lattice), 162Discriminant ideal (of a quaternion

algebra), 114Discriminant (of a binary form), 204, 221Discriminant (of a field), 39Discriminant (of a lattice), 39Discriminant (of a quadratic form), 119Discriminant (of a quaternion algebra), 147Division algebra, 79

E

Eisenstein equation, 34Eisenstein polynomial, 34Equivalent (representations), 92Equivalent (valuations), 20Euler’s function, 4Even Clifford algebra, 122Even Clifford group, 127

F

Faithful representation, 92F -algebra, 79Finite field, 3Fractional ideal, 35Free module, 11Frobenius automorphism, 73Fundamental unit, 56

G

Gauss sum, 7Genus (of a lattice), 171Genus (of a matrix), 178Genus (of ideals), 230g-ideal, 101Global field, 100Group algebra, 85

H

Hamilton quaternion algebra, 96Hasse norm theorem, 141

236 INDEX

Hasse principle, 146Hensel’s lemma, 23Hilbert reciprocity law, 141Hilbert symbol, 140

I

Ideal class, 42Ideal class group, 42Ideal group, 36Idele group, 68Idele norm, 67Imaginary archimedean prime, 66Imaginary quadratic field, 40Indecomposable (ring), 3Indefinite (quaternion algebra), 147

Inertia group, 72Integral (lattice), 161Integral (over a ring), 25, 101Integral closure, 25Integral ideal, 35Integrally closed, 25Invertible (element of a quadratic

space), 121Involution, 89Irreducible (module), 79Isotropic, 115

L

Lattice (in a real vectorspace), 50

Lattice (in a vector space), 11Left order, 101Left regular representation, 92Left semisimple, 81Level, 181Local field, 100

M

Main involution, 98Mass, 224Maximal ideal (of a valuation

ring), 17Maximal lattice, 161Maximal order, 27Minimal ideal, 81Minimal/minimum polynomial, xiMinkowski’s lemma, 52Monic polynomial, xi

N

Nonarchimedean prime, 66Nonarchimedean valuation, 16Nondegenerate quadratic space, 115

Norm (of an ideal), 35, 38Norm form, 134Normalized order function, 15

O

Order (in a quadratic extension), 213ffOrder (in an algebra), 101Order function, 15Orthogonal basis, 119Orthogonal complement, 115Orthogonal group, 116

P

p-adic field, 19p-adic integer, 19p-adic number, 19Pentagonal number, 151Prime element, 17Prime ideal, 1, 35

Primitive binary form, 204Primitive character, 7, 9Primitive root, 5Primitive solution, 203Principal genus, 230Principal ideal, 1, 35Principal ideal domain, 1Product formula, 22, 50Proper ideal, 214Pure quaternion, 134

Q

q-reduced, 180Quadratic Diophantine equation, 203Quadratic field, 40Quadratic form, 115Quadratic nonresidue (modulo a prime), 6Quadratic reciprocity law, 6Quadratic residue (modulo a prime), 6Quadratic residue symbol, 6Quadratic space, 115Quaternion algebra, 96

R

Ramification index, 28Ramified, 28, 30, 40, 57, 66, 142Real archimedean prime, 66Real character, 9Real quadratic field, 40Reciprocal, 89Reduced norm, 93Reduced trace, 93Regular representation, 47Regulator, 55

INDEX 237

Relative discriminant, 59Representation, 92Residue class degree, 28Right order, 101Right regular representation, 92Right semisimple, 81

S

Scalar extension, 87Semisimple algebra, 84Simple algebra, 81Special orthogonal group, 116Spin group, 131Spinor norm, 131Split (quadratic space), 117Split Witt decomposition, 117s-reduced, 180Strong approximation, 177, 192Subalgebra, 79Symmetry, 128

T

Tensor product (of algebras), 86

Tensor product (of fields), 47

Torsion-free (module), 11

Totally isotropic, 115

Totally ramified, 30

Triangular number, 150

Type 1, 181

Type 2, 181

U

Unique factorization domain, 1

Unramified, 28, 30, 40, 57, 66, 142

V

Valuation, 15

Valuation ring, 17

W

Weak Witt decomposition, 117

Witt decomposition, 117

Witt’s theorem, 117