Solved: Field Theory Problem: Z_p in Algebraic Closure of F

[ehrenfest]

[SOLVED] field theory problem

Homework Statement

If F is a field that has characteristic p, it must contain a copy of Z_p. Is it true that F must sit inside of the algebraic closure of Z_p? My book assumes that it does and I do not understand why?

Homework Equations

The Attempt at a Solution

 

[morphism]

Are you sure it says F is an arbitrary field of characteristic p? Because this is obviously false. For instance take the rational function field of the alg closure of Z_p - this still has characteristic p and properly contains the alg closure of Z_p.

On the other hand, this is true if F is a finite field of characteristic p.

 

[ehrenfest]

Are you sure it says F is an arbitrary field of characteristic p? Because this is obviously false. For instance take the rational function field of the alg closure of Z_p - this still has characteristic p and properly contains the alg closure of Z_p.

On the other hand, this is true if F is a finite field of characteristic p.

Sorry. F has p^r elements.

 

[morphism]

Then F is the splitting field of x^(p^r) - x over Z_p.

 

[ehrenfest]

Then F is the splitting field of x^(p^r) - x over Z_p.

I am in section 33 and splitting fields are in section 50. Thus, I do not even know what they are.

I figured it out though. F must be a finite extension of Z_p and finite extensions are always algebraic extensions and thus F must sit inside the algebraic closure of Z_p.