16
$\begingroup$

(Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory course.)

Let $K$ be a number field. Is there always a finite extension $L / K$ such that $L$ has class number 1?

If $K$ has finite class field tower (i.e. the tower of fields $(K_n)_{n \ge 0}$, where $K_0 = K$ and $K_{n+1}$ is the Hilbert class field of $K_n$, eventually terminates) then that solves the problem. But it's a well-known theorem of Golod and Shafarevich that the class field tower of $K$ doesn't terminate if $K$ is an imaginary quadratic field with enough primes ramified.

The textbook my colleague has been using claims that it follows from Golod-Shafarevich that these fields $K$ cannot be embedded in any number field with class number 1, but this implication isn't clear to me. Golod-Shafarevich shows that $K$ has no finite, solvable, everywhere-unramified extension with class number 1, but that's a much weaker statement, isn't it?

$\endgroup$
1
  • 1
    $\begingroup$ I just want to mention that this question came up in a totally different context! It was about proving the (weak) Mordell-Weil Theorem for elliptic curves over number fields. There is a proof that works only when $K$ has class number $1$. Of course, if we can embed $K$ in a field with class number $1$, then the theorem follows a fortiori in $K$ because of its truth in the larger field. $\endgroup$ Jul 9, 2012 at 5:59

1 Answer 1

16
$\begingroup$

See Proposition 1 on p.231 of Cassels and Frohlich for a proof of the claim in the textbook:

The point is that if such an $L$ exists then $K_1L$ is abelian and unramified over $L$ so it is contained in the Hilbert classfield of $L$ which is $L$ itself. By induction, this implies that $K_i \subset L$ for all $i$ so the class field tower of $K$ must be finite.

$\endgroup$
1
  • 1
    $\begingroup$ Ah, of course: the class field tower of $L$ has to be at least as big as the class field tower of $K$. Thanks! $\endgroup$ Mar 26, 2012 at 10:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.