# Show that the algebraic closure of $F$ in $K$ is an algebraic closure of $F$.

(a) Let $$K$$ be an algebraically closed field extension of $$F$$. Show that the algebraic closure of $$F$$ in $$K$$ is an algebraic closure of $$F$$.

What is algebraic closure of $$F$$ in $$K$$? The definition of algebraic closure is:

If $$K$$ is an algebraic extension of $$F$$ and is algebraically closed, then $$K$$ is said to be an algebraic closure of $$F$$

In this case, $$K$$ is more than algebraic extension so, what algebraically closed extension? I'm a little confused by this.

(b) If $$\mathbb{A} = \lbrace a \in \mathbb{C}\,|\,a\,\text{is algebraic over}\,\mathbb{Q}\rbrace$$, then, assuming that $$\mathbb{C}$$ is algebraically closed, show that $$\mathbb{A}$$ is an algebraic closure of $$\mathbb{Q}$$.

I imagine that $$\mathbb{C}$$ is an algebraically closed field extension of $$\mathbb{Q}$$ and $$\mathbb{A}$$ is the algebraic closure of $$\mathbb{Q}$$ in $$\mathbb{C}$$.

So, I would like some help to understand these definitions for can answer the two items. Thanks for the advance!

It may be useful to put some definitions:

Lema 1. If $$K$$ is a field, then the following statements are equivalente:

1. There are no algebraic extensions of $$K$$ other than $$K$$ itself.
2. There are no finite extensions of $$K$$ other than $$K$$ itself.
3. If $$L$$ is a field extension of $$K$$, then $$K = \lbrace a \in L\,|\,a\,\text{is algebraic over}\,K\rbrace$$.
4. Every $$f(x) \in K[x]$$ splits over $$K$$.
5. Every $$f(x) \in K[x]$$ has a root in $$K$$.
6. Every irreducible polynomial over $$K$$ has degree $$1$$.

Definition 1. If $$K$$ satisfies the equivalent conditions of Lema 1, then $$K$$ is said to be algebraically closed.

The algebraic closure of $F$ in $K$ denotes the set of elements in $K$ which are algebraic over $F$.
And, yes, $\mathbf C$ is an algebraically closed field (that's what D'Alembert-Gauß' theorem asserts) and it is an extension of $\mathbf Q$.
• So, and "$K$ algebraically closed extension of $F$" is just a extension of $F$ which is algebraically closed? May 3, 2018 at 23:04
• ${}$Absolutely! May 3, 2018 at 23:10