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**Extra resources for Algebraic Number Theory and Fermat's Last Theorem (3rd Edition)**

**Sample text**

An examination of the cases shows d = ~ ( p ) as This computation can be made the basis of one of the many proofs of the law of quadratic reciprocity. We shalI present the details. Then U p is a cyclic group of orderp- I . The collection of all squares of elements in U pforms a subgroup, U p 2 ,of index two. Let { & 1) = Tdenote the multiplicative group of order two. This homomorphism will be denoted by ( . We call ( u / p ) the Legendre symbol. It is usually convenient to define (alp) for a in Z to mean the value of ( .

The prime 9 # p splits as a product of two primes in Q([~(p)p]"~)if and only if ( q / p )= 1. PROOF. 6). By Proposition 9. Thusg is even if and only iffdivides ( p - l ) / 2 . Because of the characterization of the relative degree mentioned just above, this holds if and only if q(P-')/' = 1 modp. Then q has two prime factors in R if and only if ( d P ) = 1. One last computation before we reach our goal. 8 Lemma. ( - I/p) = ( - l ) ( p - ' ) / 2 for any odd primep. PROOF. (- I/p) = 1 if and only if - 1 = u 2 for some u in U p .

Q) x Gal(L,/Q). By induction, the order of Gal(Q(B)/Q) is 4 ( p " ) 4 ( n )= 4(p"n) = 4(m), which proves (a). Now let R denote the ring of algebraic integers in L,, S the algebraic integers in Q(0) and E a primitive path root of unity. Then RE&]c s and the discriminant ideal A (SIR)contains the discriminant A ( 1, E , .. , Observe that for x E L,. we have = TLp4/&). ,&'(Pa)-') = power of ( p ) . Hence A ( S / R ) 2 power ofpR. The only primes of R which can ramify in S are the divisors of p R .