We denote by Z, Q, R, and C the ring of rational integers, the fields of rational
numbers, real numbers, and complex numbers, respectively. We put
T = { z e C | |z| = l } .
We denote by Q the algebraic closure of Q in C, and for an algebraic number
field K we denote by K"ab the maximum abelian extension of K. If p is a rational
prime, Zp and Qp denote the ring of p-adic integers and the field of p-adic numbers,
For an associative ring R with identity element and an .R-module M we denote
the group of all its invertible elements and by M™ the i£-module of all
m x n-matrices with entries in M; we put
M™ for simplicity. Sometimes an
object with a superscript such as
in Section 23 is used with a different meaning,
but the distinction will be clear from the context. For x e R™ and an ideal a of
R we write x - a if all the entries of x belong to a. (There is a variation of this;
see §1.8.)
The transpose, determinant, and trace of a matrix x are denoted by
and tr(x). The zero element of R™ is denoted by 0™ or simply by 0, and the identity
element of R™ by l
or simply by 1. The size of a zero matrix block written simply
0 should be determined by the size of adjacent nonzero matrix blocks. We put
(#) = ( i ^ )
, a n d
SLn(R) = { a e GLn(R) | det(a) = 1 }
if R is commutative. If xi, ... , xr are square matrices, diag[xi, ... , xr] denotes
the matrix with
1 ? , **•f 1XJL
the diagonal blocks and 0 in all other blocks. We
shall be considering matrices x with entries in a ring with an anti-automorphism
p (complex conjugation, for example), including the identity map. We then put
x* ~
and x =
if x is square and invertible.
For a complex number or more generally for a complex matrix a we denote by
Re(a), Im(a), and a the real part, the imaginary part, and the complex conjugate
of a. For complex hermitian matrices x and y we write x y and y x if x y
is positive definite, and x y and y x if x - y is nonnegative. For r G R we
denote by [r] the largest integer r.
Given a set A, the identity map of A onto itself is denoted by id^ or 1^. To
indicate that a union X = (JiG/ Yi is disjoint, we write X = [JieI 5^. We understand
that nf=
= 1
a n d
Ef=a = 0 if a /?. For a finite set X we denote by # X or
#(X ) the number of elements in X. If H is a subgroup of a group G, we put
[G : H] = #{G/H). However we use also the symbol [K : F] for the degree
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