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On the Parity of the Class Number of the Field Q(p,q,r)

โœ Scribed by Michal Bulant

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
406 KB
Volume
68
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

If ( pร‚q)=( pร‚r)=(qร‚r)=&1, fix u pq , u pr , u qr # Z satisfying u 2 pq #pq (mod r), u 2 pr #pr (mod q), u 2 qr #qr (mod p). Then h is even if and only if (u pq ร‚r)(u pr ร‚q)(u qr ร‚p)=&1.

  1. If ( pร‚q)=1, ( pร‚r)=(qร‚r)=&1, then the parity of h is the same as the parity of the class number of the biquadratic field Q(-p, -q).

  2. If ( pร‚q)=(qร‚r)=1, ( pร‚r)=&1, then h is even.

  3. If ( pร‚q)=( pร‚r)=(qร‚r)=1, then h is even. (Moreover, if we denote by v pq , v pr , v qr , v pqr the highest exponents of 2 dividing the class number of Q(-p, -q), Q(-p, -r), Q(-q, -r), Q(-p, -q, -r), respectively, then v pqr 1+v pq +v pr +v qr .

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