Re: Related to Galois.. Is there a Galois ext of Q for each prime p with degree p?
From: Edwin Clark (eclark_at_math.usf.edu)
Date: 08/06/04
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Date: Fri, 06 Aug 2004 02:33:02 GMT
Eric wrote:
> When i use the cyclic ext of Q, I just know that there exist Galois
> ext of Q with degree p-1. But how can i do when the degree is p ?
To find an extension of Q of degree p where p is prime, first find a
prime q such that p divides q-1. Let K be an extension of degree q-1.Now
Gal(K/Q) is cyclic of order q-1 and all of its subgroups are normal. The
subgroups of Gal(K/Q) are all normal and correspond to normal
extensions of Q. If you pick the right subgroup then its fixed field
will be a normal extension of Q of degree p. [Note that by Dirichlet's
Theorem there are infinitely many primes q such that p | q - 1.]
Actually the above holds for any positive integer p > 1 and shows there
is a cyclic extension of Q of degree p.
Edwin Clark
(Everette Dade pointed this out to me in 1965.)
- Previous message: Gerry Myerson: "Re: Self-teaching of integration methods"
- In reply to: Eric: "Related to Galois.. Is there a Galois ext of Q for each prime p with degree p?"
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