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When $p$ is a prime $\equiv9\bmod16$, the class number, $h$, of $\mathbb Q(p^{1/4})$ is known to be even. In

[Charles J. Parry, A genus theory for quartic fields. Crelle's Journal 314 (1980), 40--71]

it is shown that $h/2$ is odd when 2 is not a fourth power in $\mathbb Z/p\mathbb Z$. Does this still hold when 2 is a fourth power?

Some years ago I gave an (unpublished) proof that this is true provided the elliptic curve $y^2=x^3-px$ has positive rank, and in particular that it is true on the B. Sw.-D. hypothesis. It's known that the above curve has positive rank for primes that $\equiv5$ or $7\bmod16$, but to my knowledge $p\equiv9\bmod16$ remains untouched. But perhaps there's an elliptic-curve free approach to my question?

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5  
PARI confirms h has just a single 2 in it for your $p$ up to 30,000. (That's just 397 primes, however.) –  KConrad Jul 28 '10 at 23:02
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For those interested--I've written out the elegant argument that Franz gave in more detail. See: A theorem of Lemmermeyer on class numbers, arXiv AC 1009.3990 –  paul Monsky Sep 25 '10 at 14:43

1 Answer 1

up vote 13 down vote accepted

Let $p \equiv 1 \bmod 8$ be a prime number, let $K = {\mathbb Q}(\sqrt[4]{p})$, and let $F$ be the quartic subfield of the field of $p$-th roots of unity. An easy exercise involving Abhyankar's Lemma shows that $FK/K$ is an unramified quadratic extension, hence the class number of $K$ is always even.

The field $KF$ has the quartic subfield $L = {\mathbb Q}(\sqrt{u})$, where $u$ is the fundamental unit of $k = {\mathbb Q}(\sqrt{p})$. An routine application of the ambiguous class number formula to $L/k$ shows that $L$ has odd class number (there are two ramified primes, one infinite and the other one above $2$; clearly $-1$ is not a norm residue at the infinite prime).

Now I claim that if $p \equiv 9 \bmod 16$, the class number of $KF$ is odd. By class field theory, this implies that the $2$-class number of $K$ must be $2$. An application of the ambiguous class number formula to $KF/L$ shows that the $2$-part of the ambiguous class group has order $$ h = \frac{2}{(E:H)}, $$ where $E$ is the unit group of $L$ and $H$ its subgroup of units that are norms from all completions of $KF$: in fact, only the two prime ideals above $p$ are ramified in $KF/L$. Thus it is sufficient to show that $E \ne H$. I will show that $\sqrt{u}$ is a quadratic nonresidue modulo the primes $\mathfrak p$ above $p$. But if $u = T + U \sqrt{p}$ (replace $u$ by $u^3$ in order to guarantee that $T$ and $U$ are integers), then $(\sqrt{u}/{\mathfrak p})_2 = (u/\mathfrak p)_4 = (T/p)_4 = (T^2/p)_8 = (-1/p)_8 = -1$ because $p \equiv 9 \bmod 16$; here we have used the congruence $T^2 \equiv -1 \bmod p$.

The reason why the case $(2/p)_4 = -1$ is easier is because in this case, the ideal above $2$ ramified in $K$ generates a class of order $2$ in the $2$-class group, whereas this prime generates a class with odd order if $(2/p)_4 = +1$, which means that there is no strongly ambiguous ideal class in this case.

Edit. Paul Monsky has kindly written up this argument, filled in all the details, and made it available here. Thanks!

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Thanks for settling this, Franz. The idea of showing that a degree 2 extension of K had odd class number never occurred to me, and my elliptic-curve semi-proof lay on a shelf for 20 years. I'll send a copy of it to you anyway. By the ambiguous class number formula I take it you mean Lemma 4.1 of Lang's Cyclotomic II? I had trouble at first seeing why L is contained in KF, but with that clear to me now, I think I can follow your argument. –  paul Monsky Aug 3 '10 at 20:28

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