Let $K,L$ over $Q$ be field extension of prime degrees. Prove if $[KL:Q]<[K:Q][L:Q]$, then the Galois closure of $K/Q$ is the Galois closure of $L/Q$.

Asked Feb 28 '24 at 13:20

Active Mar 08 '24 at 08:44

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Let $K$ and $L$ over $Q$ be field extension of prime degrees. Prove that if $[KL:Q]<[K:Q][L:Q]$, then the Galois closure of $K/Q$ equals to the Galois closure of $L/Q$.

I know the two extensions must be simple extensions, and the two degrees are equal. But I cannot figure out the rest.

edited Mar 08 '24 at 08:44

Shaun

47,747

asked Feb 28 '24 at 13:20

z2014430

I am intrigued by this question. It is, indeed, easy to show that the two primes are equal. It is easy to find examples of such pairs of fields $K,L$ for all primes $p$. And in all the examples I have thought of $K$ and $L$ are conjugates implying the claim. Yet I don't see why they would need to always be conjugates. – Jyrki Lahtonen Mar 05 '24 at 05:37
Somewhat related. – Jyrki Lahtonen Mar 05 '24 at 05:37
How did you run into this question? – Jyrki Lahtonen Mar 05 '24 at 07:11
The main claim follows from Derek Holt's answer to my related question. – Jyrki Lahtonen Mar 08 '24 at 11:00

Let $K,L$ over $Q$ be field extension of prime degrees. Prove if $[KL:Q]<[K:Q][L:Q]$, then the Galois closure of $K/Q$ is the Galois closure of $L/Q$.

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