Galois theory question

[Grant Sedgwick]

Can anyone please explain why the following statement is true?
Unfortunately the text omitted a proof.

If K and L are fields where L is a finite extension of K, then there exists
a polynomial f(x) in K[x] so that L is contained in the splitting field of
f(x) over K.

Many thanks.

[Arturo Magidin]

Since L is a finite extension of K, there is a finite basis for L over
K, a1,...,an.

let f_i(x) be the minimal polynoomial of a_i over K. Let f(x) = f1(x)
*...*fn(x). Then the splitting field of f(x) over K certainly contains
a1,...,an, hence contains L.

There are, of course, other ways of doing this, but this particular
sledgehammer is sure to work.

--
Arturo Magidin

[Grant Sedgwick]

Great! Cheers.

"Arturo Magidin" <mag...@member.ams.org> wrote in message
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