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

In summary, the conversation discusses whether or not a field with characteristic p must be contained within the algebraic closure of Z_p. It is determined that this is true if F is a finite field of characteristic p, as finite extensions are always algebraic extensions and thus F must sit inside the algebraic closure of Z_p.
  • #1
ehrenfest
2,020
1
[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

 
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  • #2
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.
 
  • #3
morphism said:
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.
 
  • #4
Then F is the splitting field of x^(p^r) - x over Z_p.
 
  • #5
morphism said:
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.
 

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

1. What is a field theory problem?

A field theory problem is a mathematical problem that involves studying the properties and structure of fields, which are algebraic structures that have addition, subtraction, multiplication, and division operations defined on them. Examples of fields include the rational numbers, real numbers, and complex numbers.

2. What is Z_p in algebraic closure of F?

Z_p refers to the set of integers modulo a prime number p. In algebraic closure of F, Z_p is used as the base field for constructing a larger field that contains all algebraic numbers over F. This larger field is called the algebraic closure of F.

3. How is Z_p related to field theory?

Z_p is related to field theory in the sense that it is a field itself and can be used as a building block for constructing larger fields. It is also used to study the properties of fields, such as their algebraic closure, and to prove theorems in field theory.

4. What are some applications of field theory?

Field theory has many applications in mathematics and other fields such as physics and engineering. Some examples include Galois theory, which has applications in cryptography and coding theory, and differential Galois theory, which is used in the study of differential equations. Field theory is also used in algebraic geometry and algebraic number theory.

5. Are there any unsolved problems in field theory?

Yes, there are many unsolved problems in field theory, as in any other area of mathematics. Some of these include the inverse Galois problem, which asks whether every finite group can be realized as the Galois group of a finite extension of the rational numbers, and the Langlands program, which aims to unify various areas of mathematics using the concept of automorphic forms and their associated L-functions.

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