By Takashi Ono (auth.)

ISBN-10: 0306434369

ISBN-13: 9780306434365

ISBN-10: 146130573X

ISBN-13: 9781461305736

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Therefore, we get [G(K/k)] = [K: k]. Now we shall introduce some notations for a finite Galois extension K / k. We denote by [ff the set of all fields F such that K -:::J F -:::J k, and by Cfj the set of all subgroups G of G(K/k). Next, we define a mapping #: [ff~ Cfj and a mapping b: Cfj~ [ff as follows: F# = {aE G(K/k); a(a) = a, Va E F}, G b = {a E K; a(a) = a, Va E G}, for FE :IF, for G E

7L is called the ring PROOF. We must verify that a, f3 E 7L::} a ± f3, af3 E 7L. Obviously, we can assume that both of a and f3 are not zero. Suppose a satisfies ai E n :2: 1, m :2: 1. Z, and f3 satisfies bi E Z, Then consider the nonzero module M: M= . {r = ~ Cij~f3j; Cij E Z, 0 s; is; n - 1, 0 s; j s; m - I}. ,J In view of the relations for a and f3, we see that aM c M and f3M eM. D. 20. 23. (J) is a subfield of the field c. (J) is called the field of algebraic numbers or the algebraic closure of Q.

Furthermore, solve the following three equations: 7x == 5 (13), x 2 == -1 (13), x 3 == 5 (13). 11. 18. A complex number a- E C is an algebraic number if there is a polynomial f(X) E Q[X], degf 2::: 1, such that f(a-) = O. We denote by 0 the set of all algebraic numbers. 23. Since a- E Q is a root of f(X) = X - a- E Q[X] we see that Q c 0. Vi is not in Q but is in 0 because it is a root of f(X) = X 2 - 2 E Q[X]. yCI is a root of f(X) = X 2 + 1 E Q[X], hence it is algebraic. For m 2::: 1, a- = e 2 :J

### An Introduction to Algebraic Number Theory by Takashi Ono (auth.)

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