If you have fields with algebraic over , show that,

is the minimal field containing and .

(Note: -a simple extension).

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- May 21st 2006, 10:29 AMThePerfectHackerIs Simple Extension Minimal?
If you have fields with algebraic over , show that,

is the minimal field containing and .

(Note: -a simple extension). - May 21st 2006, 11:59 AMrgep
The map given by evaluation at is a ring homomorphism and hence its image, which you denote , is a subring of

*E*. All you need to show is that it is closed under taking multiplicative inverses. Note for use in a moment that a subring of a field is an integral domain. The fact that is algebraic is equivalent to saying that is a finite-dimensional*F*-vector space. Let be a non-zero element of . Multiplication by is an*F*-linear endomorphism and has trivial kernel (integral domain), hence is surjective by finiteness of the dimension and the rank--nullity formula. So 1 is in the image, hence a multiple of , which is thus invertible. - May 21st 2006, 02:29 PMThePerfectHacker
I was thinking about this today, maybe you can do this?

You have with algebraic over . Let there exist a field with such as,

,

Any element can be expressed as,

with because every finite dimensional vector basis has a finite basis. Therefore thus, and thus, .

Am I missing something? - May 21st 2006, 02:34 PMrgep
I realise that I didn't actually address the issue of being the

*minimal*field, just that it actually was a field. But any field containing clearly contains any polynomial in . Hence the minimality. - May 21st 2006, 02:36 PMThePerfectHackerQuote:

Originally Posted by**rgep**

- May 21st 2006, 10:56 PMrgep
Any field that contains

*F*and contains must contain any polynomial in with coefficients in*F*. The previous part of my argument showed that the ring comprised of polynomials in with coefficients in*F*is actually a field. Hence it is the minimal subfield of*E*containing both*F*and . - May 22nd 2006, 03:12 PMThePerfectHackerQuote:

Originally Posted by**rgep**

- May 23rd 2006, 03:47 AMDMT
Isn't this all circular? I thought was defined to be the smallest the field containing both F and .

- May 23rd 2006, 02:17 PMThePerfectHackerQuote:

Originally Posted by**DMT**