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INDEX 271
linear map, 227
linearization, 228
µ (M
¨
obius function), 197
mathematical induction, 97
matrix, 114–123
entry, 117
inverse, 126
invertible, 126–134
multiplication, 117–120
zero, 145
Mazur, Barry, 43, 231, 249
Merel, Lo
¨
ıc, 255
Mertens Conjecture, 197
modular arithmetic, 31–41
modular deformation, 252
modular form, 236, 238
Fourier coefﬁcients, 236
level, 237, 246
weight, 246
modulo, 33
modulus, 33
morphism, 7
surjective, 159
N (norm), 183
n-dimensional vector, 226
neutral element, 2
Newton, Isaac, 52, 73
norm, 183
number
algebraic, 89
complex, 42–48
imaginary part, 44
real part, 44
composite, 34
irrational, 52
prime, 34
rational, 52
real, 43
system, 36
O (point at inﬁnity), 106
O
K
, 189
Odlyzko, Andrew, 198
one-to-one correspondence, 5
% (remainder), 38
permutation, 22, 92
Poincar
´
e Conjecture, 197
polynomial
degree, 68, 88
irreducible, 186
minimal, 181, 184
root, 61, 88
product ideal, 189
Q (rational numbers), 52, 53
Q
alg
, 48, 89, 91, 92, 95, 96, 98, 100,
101, 149, 160, 180, 181, 189,
193, 228
Q(f ), 150, 178
R (real numbers), 43, 53
Ramanujan’s τ function, 238
ramiﬁed, 179, 194
reciprocity law, 79, 239–240
relatively prime, 131
representation, 3–9
absolutely irreducible, 248
equivalent, 173
faithful, 137, 146, 168–169, 171,
175
Galois, 194, 196
modular, 252
group, 135–148
image, 263
irreducible, 248
linear, 115, 139, 146–148, 164, 226
matrix, 139, 164
one-dimensional, 204–205
permutation, 138, 160
trivial, 148, 170
two-dimensional linear Galois, 206
restriction morphism, 157
Riemann Hypothesis, 197, 198, 260

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