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Let $K$ be an algebraic number field. Let $h$ be the class number of $K$. Let $l$ be a prime number. Suppose $l$ does not divide $h$. Let $L/K$ be a Galois extension of degree $l$. Let $H$ be the Hilbert class field of $K$. Suppose $L/K$ is unramified. Since $L/K$ is abelian, $L$ is a subfield of $H$. Hence $l$ divides $h$. This is a contradiction. Hence $L/K$ is ramified.

My Question Can we prove that $L/K$ is ramified without using class field theory? In other words, can we prove the following proposition without using class field theory?

Proposition Let $K$ be an algebraic number field. Let $h$ be the class number of $K$. Let $l$ be a prime number. Suppose $l$ does not divide $h$. Let $L/K$ be a Galois extension of degree $l$. Then $L/K$ is ramified.

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  • $\begingroup$ If you’re asking for a proof of a proposition without use of CFT, I’d like a precise statement of the proposition. $\endgroup$ – Lubin Sep 21 '13 at 2:57
  • $\begingroup$ @Lubin I added the precise statement to my question. Regards, $\endgroup$ – Makoto Kato Sep 21 '13 at 5:35
  • $\begingroup$ OK, thanks, that’s what I thought you might be asking. At first glance, it certainly does look fundamentally class-field-theoretical... $\endgroup$ – Lubin Sep 21 '13 at 13:43
  • $\begingroup$ @Lubin It seems that Kummer proved the above proposition when $l$ is odd and $K$ is the $l$-th cyclotomic field. $\endgroup$ – Makoto Kato Sep 21 '13 at 22:11

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