By Takashi Ono (auth.)

ISBN-10: 0306434369

ISBN-13: 9780306434365

ISBN-10: 146130573X

ISBN-13: 9781461305736

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**Extra resources for An Introduction to Algebraic Number Theory**

**Sample text**

A polynomial P = P(X) in K[X] is an expression ai E K, = deg P. K[X] forms a ring with respect to addition and multiplication defined in a natural way; the field K is a subring of the ring K[X] and 1 E K serves as the unit element for K[X]. For P, Q E K[X], with P =1= 0, Q =1= 0, we have deg(PQ) = deg(P) + deg(Q). The ring K[X] resembles the ring 7L. In what follows, we shall asterisk the number for a statement on K[X] corresponding to a similar statement on 7L mentioned earlier. 37. Prove that (K[X]Y = K X for a field K.

Prove that DKlk(a) = (_1t(n-l)/2NKlkf'(a) where f~ denotes the derivative of fo:. 8. Let Kio. 4). Verify that DK10(Ym) = 4m. 9. 5). 4. 13 we used the Gauss sum (for the Legendre character) to prove the Gauss reciprocity law. 1171.. be the finite field with 1 elements and X be a character of the multiplicative group 1Ft. We shall denote by E the identity character: E(X) = 1, Vx E 1Ft. 12, the Legendre character Al is the unique character of order 2 (in the group ~t of all characters of 1Ft). We shall put X(O) = 0 for any X and occasionally consider X as a function defined on the whole set IF/.

Then the mapping #: [ff~ Cfj, b: Cfj~ [ff are both bijective, order reversing, and inverse of each other: b . # = id,'9'", # . b = id'§. pKj Gb k {r~# =t G G(K/k) 51 Concepts of Algebraic Number Fields PROOF. 7) and E

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

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