Kronecker-Jacobi symbol and Quadratic Reciprocity. Q b /Q p

Transcription

1 Kronecker-Jcoi symol nd Qudrtic Recirocity Let Q e the field of rtionl numers, nd let Q, 0. For ositive rime integer, the Artin symol Q /Q hs the vlue 1 if Q is the slitting field of in Q, 0 if is rmified in Q, nd 1 otherwise i.e., if Q Q nd is inertil. Here we hve identified the Glois grou Aut Q with sugrou of the multilictive grou {±1}. For ritrry ositive rtionl m i1 n i i, we set Q /Q : m i1 ni Q /Q : 1 if m 0, i.e., 1. i This gives rise to homomorhism recirocity m from the multilictive grou of non zero rtionl numers > 0 reltively rime to the discriminnt d of Q /Q, into the Glois grou of Q /Q. Exercise. If div is the norm of divisor of Q, nd,d 1, then lies in the kernel of the recirocity m. For ny 0 x Q, the sign of x is sgnx: x/ x 1 εx where εx: sgnx 1/. The numer x cn e written uniquely in the form x x y, where x 1 εx 1... m with m 0 distinct rimes i. For nonzero rtionl,, define the Kronecker symol Q /Q : 1 εε 1

2 i ii iii iv One checks tht: 1 1; 1 ε sgn. 1 1 { d if 1 mod 4. Recll: d d 4 otherwise 0 iff nd d re reltively rime. s long s the right hnd memer does not vnish. 1 1 Well known fcts out ehvior of rimes in qudrtic numer fields give, further: v If is n odd rime nd is n integer then is just the usul Legendre symol. vi If does not divide d i.e., d 1 mod 4, then ±1 ccording s is or is not squre mod d 8. Now iv, v, vi llow us to define solely in terms of Legendre symols, to wit: 1 εε rime d This is how it ws done originlly. From this definition, one gets t once: 1 1 vii s long s the right hnd memer does not vnish. viii If > 0 nd 1, re integers with 1 mod d, then 1 If is n odd ositive integer, it is even sufficient tht 1 mod

3 The hert of the recirocity lw lies in the following fct. Theorem. The ming induces homomorhism χ from the multilictive grou Z/dZ of units in Z/dZ onto the Glois grou Aut Q. This χ, clled the qudrtic chrcter of Q /Q when Q Q i.e., 1, is the unique homomorhism tking ny odd rime not dividing to the Legendre symol. In other words: if 1 mod d then 1. Moreover, if Q Q then there exists with 1. Proof. Uniqueness is shown y relcing y [n] : + nd where n is such tht [n] is ositive nd odd, nd then fctoring [n] into rimes. Such n n clerly exists if is odd or if is even nd reltively rime to d so tht d is odd. It is n exercise to show tht the unique qudrtic numer field with discriminnt d nmely Q d is sufield of the cyclotomic field Qζ d, where ζ d is rimitive d -th root of unity. [Strt with the fcts tht Q ± Qζ 8 nd tht for n odd rime, Q Qζ.] For ny rime, is the imge of under the Artin m into Q, hence the restriction of the imge of under the Artin m into Qζ d d : d, i.e., the utomorhism tking ζ d to ζ d. Here, in view of iv, we cn relce y ny ositive integer ; nd furthermore, to do the sme for negtive, it will suffice to do it for 1, i.e., to show tht the utomorhism θ tking ζ d to ζ 1 d tkes to 1 sgn. But this follows t once from the fct tht the fixed field of θ is Q ζ d R whenever d > 1. Surjectivity of χ results from its fctoriztion s Z/dZ Aut Qζ d Aut Q. Corollry 1. For ny odd integer, 1 sgn ±1 ccording s is or is not squre mod d 1 4. Q.E.D. Proof. Since d 1 4, the Theorem with 1 reduces the rolem to the two simle cses 1, 3. 3

4 Corollry. For ny odd integer, ±1 ccording s is or is not squre mod d 8. Proof. Since d 8, the Theorem with reduces the rolem to the simle cses 1,3,5,7. Corollry 3. If q is n odd rime, q 1 q 1 q, nd 0 is n integer, then q q q Proof. Note tht dq q. So for vrile with,q 1, nd re q oth homomorhisms of the cyclic grou of units in Z/qZ onto grou of order. But there is only one such homomorhism, hence the ssertion holds in this cse. Corollry 4. Let e ny odd integer, set : 1 1 so tht d, nd let 0 e n integer. Then Proof. We my ssume tht is squre free, nd, 1; nd then use 1 1 nd to reduce, vi iv nd vii, to Corollry 3. Comining these corollries we otin the recirocity lw for the Kronecker symol: If,d,d 1, then, with m 0, n 0, 0 nd 0 odd, it holds tht εε Proof. We cn relce y nd y see ii, i.e., we my ssume tht nd re squrefree integers. Then t lest one of, must e odd; we my ssume odd. Now if 1 mod 4 then 0 1 is even, 0, nd y Corollry 4, whence the ssertion in this cse. 1 εε, 4

5 If 3 mod 4 then 0, 0 since 1,d,4, nd y vii, vi nd Corollry 4, εε, whence the ssertion in this cse too. Q.E.D. Remrk. The kernel of χ consists of ll residue clsses in Z/dZ of norms of idels in the ring of integers of Q which re reltively rime to d. Sufficiency follows from the exercise on ge 1. When is rime, nd χ : 1 then slits in Q, so is norm. Then use Dirichlet s theorem on rimes in rithmetic rogressions to see tht for ny integer with,d 1, there exists rime such tht mod d. Exmle is rime numer. Is 819 qudrtic residue or non residue? 819/ / /819 /819679/ / / / /103 61/ /61 4/61 /611/61 61/1 19/1 1/19 /19 1 nonresidue. Exercises. 1. Check tht 5 6 1, nd tht 5 is not squre mod 6.. Show tht 50009/ is rime. 3. Try to show, without using the Theorem, tht for integers, with 0 < < d, d Remrks. 1. The key to the ove roch to recirocity ws the fct tht ny qudrtic extension of Q is contined in cyclotomic field. An imortnt theorem Kronecker Weer sttes tht ny elin extension of Q is contined in cyclotomic field. It follows, s in the ove roof, tht if K/Q is n elin extension, with, sy K Q n 1 then the slitting field of rime which does not rmify in K deends only on the residue clss of in Z/nZ. Similr simle decomosition lws hold for elin extensions of ritrry numer fields; this is sic fct of clss field theory.. In more sohisticted tretments of recirocity, sign comlictions re delt with more elegntly in terms of ehvior t the infinite rime. 5

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