# Given an algebraic alpha be of degree n over F, show at most . . .

1. Jul 8, 2007

### barbiemathgurl

let alpha be algebraic over F of degree n, show that there exists at most n isomorphisms mapping F(alpha) onto a subfield of bar F (this means the algebraic closure).

thanx

2. Jul 8, 2007

### StatusX

Let phi be one of the isomorphisms, and apply phi to the minimal polynomial for alpha. This should give you a condition on phi(alpha).

3. Jul 8, 2007

### Kummer

If $$\phi:F(a) \mapsto E$$ is an isomorphism for $$E\leq \bar F$$ then $$a\mapsto b$$ where $$a\mbox{ and }b$$ are conjugates. Conversely, if $$a\mapsto b$$ and their are conjugates then by the Conjugation Isomorphism Theorem there is exactly one such isomorphism leaving $$F$$ fixed. Let the irreducible monic polynomial for $$a$$ be $$c_0+c_1x+...+c_nx^n$$ then there are at most $$n$$ zeros, and hence at most $$n$$ conjugate elements.