THE QUADRATIC AND QUARTIC CHARACTER OF CERTAIN QUADRATIC UNITS I PHILIP A. LEONARD AND KENNETH S. WILLIAMS

Transcription

1 PACFC JOURNAL OF MATHEMATCS Vl. 7, N., 977 THE QUADRATC AND QUARTC CHARACTER OF CERTAN QUADRATC UNTS PHLP A. LEONARD AND KENNETH S. WLLAMS Let ε m dente the fundamental unit f the real quadratic field Q(Vm). t is ur purpse t evaluate the ratinal quadratic and biquadratic residue symbls f ε m mdul a prime p fr certain values f m. We use the ntatin (εjp) and (ε m /p) 4 thrughut this paper as ratinal quadratic and biquadratic residue symbls, interpreting Vm as an integer mdul p. n 969 Barrucand and Chn [] prved, using the arithmetic f Q(y r^ 9 s that p = c 2 + 8d 2, then " ( L p τ/~2~), that if p = 8n + is prime, Since then a number f similar results have been btained fr certain ther quadratic and quartic symbls using such tls as cycltmy, ratinal biquadratic reciprcity laws, etc. (see Brandler [2], Lehmer [4], [5], [6]). n this paper we apply the ideas f Barrucand and Chn [] in ther biquadratic fields with unique factrizatin, thereby reprving sme knwn results, prving sme cnjectures f E. Lehmer [6] and btaining sme additinal new results. The methd succeeds in the 2 imaginary bicyclic biquadratic fields having class number and which cntain QiV ϊ), QCl/^) r ζ>(τ/~2) as a subfield. t wuld be interesting t knw if similar techniques can be used in the remaining 26 imaginary bicyclic biquadratic fields with class number r t determine ctic symbls. (Fr a cmplete list f the imaginary bicyclic biquadratic fields with class number see Brwn and Parry [3].) We nw sketch the methd used. First the quadratic r quartic symbl under cnsideratin is expressed in terms f the representatin f p by the indefinite frm assciated with the real quadratic subfield f the biquadratic field. This is accmplished using Jacbi's frm f the law f quadratic reciprcity, and the results are given in the table belw. n the case f thse results invlving quartic symbls it is first necessary t bserve that 2e m is a square in the quadratic subfield, and this brings in the symbl (2/p) 4 whse value is well knwn, viz., if p = 8n + is prime s that p a 2 + 6b 2 = c 2 + 8d 2 then (2/p) 4 = (-) 6 = (-l) +d. Next we cnsider a prime

2 2 PHLP A. LEONARD AND KENNETH S. WLLAMS i 2k + <? P A u φ j *Φ cβ u (< 5δ + J dd, 5> u S 8. f i ί % J A A * Si, 6 * 7 <M». T^ <M + CO >

3 THE QUADRATC AND QUARTC CHARACTER 3 l ε 4 S3 β* 5" 3 ih rh a 2 ϋ ϋ.* 7 Jfl* "β -* 3 s + rh "5 + J «t3 ^^ C3 w O w <N c _ * J > > ' as ]

4 4 PHLP A. LEONARD AND KENNETH S. WLLAMS factr f p in the biquadratic field under cnsideratin and by cmputing partial nrms we btain representatins f p by the three quadratic frms assciated with the three quadratic subfields. This infrmatin is then used t derive apprpriate cngruence relatins between the representatins. Finally this infrmatin is cmbined t btain the quadratic r quartic symbl slely in terms f representatins by the psitive definite quadratic frms. f ε m has nrm- we require ( /p) + in rder that (εjp) be unambiguusly defined. Our results are given in the accmpanying table. As the methd f prf is the same fr each field we just give the details fr QO/^lZ, V-ΐ). We have 2ε 4 = 2(5 + 4VΊ4) = (4 + l/ϊϊ) 2, s that (ejp) =. Next as u s ±2λ/ΐiv (mdp) with u, v > we have /4 + T/4\ = /2v\/u + 8tA \ p ) \p A p ) as (A) = = ( - ) (by Jaebi's law) u = ( Sv * ) (as p = Sv 2 (md u + Sv)) Nw let 7r be a prime factr f p in Q{V 2, V 7) s that there are integers A, B, C, D such that with A = B (md 2), C =Z> (md 2), see fr example [7]. Frming relative nrms f π in the three quadratic subfields f Q(i/ 2, τ/^7) (as in []) we can specify: (c, rf) = f (A 2 + 7J5 2-2C 2 - UD 2 ), (AC - 7BD) \4 4

5 THE QUADRATC AND QUARTC CHARACTER 5 (u, v) = (±-(A* + 7B 2 + 2C 2 + UD 2 ), λ(ad + BC)), \4 4 / (, m ) = (^(A C 2-4D 2 ), AB - 2CD)), where ϊ and m are integers such that V + 7m 2 = 4p, ΪΞ m(md 2). Clearly ϊ and m are nt bth dd s that A = B= (md 2). Hence x = (A 2 - B 2 + 2C 2 - UD 2 ), y = ±(AB - 2CD), and SOCΞDΞO (md 2). Setting A = 2A 9 B = 2B 9 C = 2C 9 D = 2D, we btain w = A\ + 75? + 2C\ + UD!, Hence as w is dd exactly ne f A ίf B is even. We just treat the case A t even, B γ dd, say A = 2A 2, B x = 2J5 2 +, as the ther case is exactly similar. We have u^aat + Ί + 2C 2 ί + 6D 2 (md 8), d = D (md 2), y Ξ= A 2 + da (md 2). t nw easily fllws frm the fllwing table that d + y == (md 2) if and nly if u ΞΞ ± (md 8). A 2 (md 2) C^md 2) A(md 2) c? + y{mά 2) u(md 8)

6 6 PHLP A. LEONARD AND KENNETH S. WLLAMS Since (A) = (_!)." and (A) ^ ( +, if u S ± (md8), x P h X u / (-, if w Ξ ±3 (md8), the prf f the result is cmplete. n sme f the ther fields certain cmplexities arse. Fr example if either Q(V r^ϊ) r Q(τ/ 3) is ne f the subfields care had t be taken in identifying the slutins f the crrespnding representatin because f the presence f units Φ ±. Whenever the questin f whether p r 4p is represented by the apprpriate frm there was an increase in the number f cases t be cnsidered. n thse cases in which the parity f n is needed it was handled by relating it t an apprpriate representatin, fr example if p = α = Sn + then n = (a 2 - l)/8 (md 2) was used. Finally we mentin that whenever the number f cases t cnsider became excessive we used Carletn University's Sigma 9 cmputer t treat them. n a frthcming paper we will discuss generalizatins f these results as well as ther results f a similar nature. REFERENCES. P. Barrucand and H. Chn, Nte n primes f the type x 2 +32y 2, class number, and residuacity, J. reine angew. Math., 238 (969), J. Brandler, Residuacity prperties f real quadratic units, J. Number Thery, 5 (973), E. Brwn and C. J. Parry, The imaginary bicyclic biquadratic fields with class number, J. reine angew. Math., 266 (974), E. Lehmer, On the quadratic character f quadratic surds, J. reine angew. Math., 22 (97), # f Q n sme special quartic reciprcity laws, Acta Arith., 2 (972), Q t f Q n the quartic character f quadratic units, J. reine angew. Math., 268/269 (974), K. S. Williams, ntegers f biquadratic fields, Canad. Math. Bull., 3 (97), Received March, 976 and in revised frm July 6, 976. Research by the secnd authr was supprted by Natinal Research Cuncil f Canada Grant N. A ARZONA STATE UNVERSTY AND CARLETON UNVERSTY OTTAWA, ONTARO, CANADA