# Thread: Is Simple Extension Minimal?

1. ## Is Simple Extension Minimal?

If you have fields $F\leq E$ with $\alpha \in E$ algebraic over $F$, show that,
$F(\alpha)$ is the minimal field containing $F$ and $\alpha$.

(Note: $F(\alpha)=\phi_{\alpha}[F[x]]$-a simple extension).

2. The map $\phi_\alpha : F[X] \rightarrow E$ given by evaluation at $\alpha$ is a ring homomorphism and hence its image, which you denote $F(\alpha)$, 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 $\alpha$ is algebraic is equivalent to saying that $F(\alpha)$ is a finite-dimensional F-vector space. Let $\beta$ be a non-zero element of $F(\alpha)$. Multiplication by $\beta$ 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 $\beta$, which is thus invertible.

You have $F\leq E$ with $\alpha$ algebraic over $F$. Let there exist a field $K$ with $\alpha \in K$ such as,
$F\leq K\leq F(\alpha)\leq E$,
Any element $\beta \in F(\alpha)$ can be expressed as,
$\beta =a_0+a_1\alpha +...+ a_{n-1}\alpha ^{n-1}$ with $a_i \in K$ because every finite dimensional vector basis has a finite basis. Therefore $\beta \in K$ thus, $K\leq F$ and $F\leq K$ thus, $K=F$.

Am I missing something?

4. 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 $\alpha$ clearly contains any polynomial in $\alpha$. Hence the minimality.

5. Originally Posted by rgep
$\alpha$ clearly contains any polynomial in $\alpha$. Hence the minimality.
What do you mean by that?

6. Any field that contains F and contains $\alpha$ must contain any polynomial in $\alpha$ with coefficients in F. The previous part of my argument showed that the ring comprised of polynomials in $\alpha$ with coefficients in F is actually a field. Hence it is the minimal subfield of E containing both F and $\alpha$.

7. Originally Posted by rgep
Any field that contains F and contains $\alpha$ must contain any polynomial in $\alpha$ with coefficients in F. The previous part of my argument showed that the ring comprised of polynomials in $\alpha$ with coefficients in F is actually a field. Hence it is the minimal subfield of E containing both F and $\alpha$.
You mean a polynomial F[x] evaluated at $\alpha$.

8. Isn't this all circular? I thought $F(\alpha)$ was defined to be the smallest the field containing both F and $\alpha$.

9. Originally Posted by DMT
Isn't this all circular? I thought $F(\alpha)$ was defined to be the smallest the field containing both F and $\alpha$.
Maybe in some texts. The version I seen is that F(a) is defined as above.