Ring Theory: Show Phi(a)= a^p is Isomorphism

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In summary, the exercise is asking to prove that the mapping phi:R-->R defined by phi(a)= a^p is a isomorphism when given a commutative ring R with a prime characteristic p. This involves using Fermat's little theorem, specifically considering \phi(a+b) \mod p.
  • #1
ECmathstudent
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Homework Statement


Given a commutative ring R with a prime characteristic p, show that the mapping phi:R-->R defined by phi(a)= a^p is a isomorphism

Homework Equations


Fermat's little theorem(I think)

The Attempt at a Solution



I'm pretty sure Fermat's theorem must have something to do with this. Or I could be completely wrong, it just popped into my head very quickly and I haven't really been able to find out if it applies in this case (some sort of generalization, at least), how to show it applies, or come up with any other idea.

Also, sorry for the formatting, I don't know LaTeX
 
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  • #2
You will want to consider [tex]\phi(a+b) \mod p[/tex].
 
  • #3
roight
 

What is Ring Theory?

Ring Theory is a branch of abstract algebra that studies the properties of mathematical structures called rings. A ring is a set of elements with two binary operations, usually addition and multiplication, that satisfy certain algebraic properties.

What is an isomorphism?

An isomorphism is a bijective function between two mathematical structures that preserves the structure and operations of the structures. In the context of Ring Theory, an isomorphism between two rings means that the two rings have the same algebraic properties and can essentially be considered the same structure.

What does it mean for Phi(a) to equal a^p?

In this context, Phi(a) refers to the image of an element a in a ring under the function Phi. This function is defined as taking an element and raising it to the power of p. Therefore, Phi(a) = a^p means that the image of a under the function Phi is equal to a raised to the power of p.

How is the isomorphism between Phi(a) and a^p shown?

To show that Phi(a) = a^p is an isomorphism, we need to prove that it is a bijective function and that it preserves the algebraic properties of the rings. This can be done by showing that the function is one-to-one and onto, and that it preserves addition, multiplication, and the identity element.

Why is this isomorphism important in Ring Theory?

This isomorphism is important because it allows us to understand how certain algebraic properties are preserved under the function Phi. It also allows us to prove theorems and make connections between different rings, which can help us to better understand the structures and properties of rings in general.

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