- #1

ehrenfest

- 2,020

- 1

**[SOLVED] extensions fields**

## Homework Statement

Can someone help me with these true or false problems:

1) Every algebraic extension is a finite extension.

2)[tex]\mathbb{C}[/tex] is algebraically closed in [tex]\mathbb{C}(x)[/tex], where x is an indeterminate

3)[tex]\mathbb{C}(x)[/tex] is algebraically closed, where x is an indeterminate

4)An algebraically closed field must be of characteristic 0

Recall that an extension field E of a field F is a finite extension if the vector space E over F has finite dimension.

## Homework Equations

## The Attempt at a Solution

1) the converse is true (it was a theorem in my book). this direction is probably not true (or else it would have also been a theorem in my book). But I need a counterexample

2) I am confused about the notation. My book has always denoted the ring of polynomials of a field f as F[x], never F(x). Furthermore, when a is an element of an extension field of F, then F(a) means F adjoined to a. But I have absolutely no idea what this means when x is an indeterminate?

3) same as 2

4) A field of characteristic 0 must contain a copy of the rationals. Does that help?