Searching for Closure: Mult. Inverses & Addition

[stripes]

Homework Statement

Homework Equations

The Attempt at a Solution

Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a^-1 is also in G.

So a*a^-1 = 1_F, but is a^-1 in G...so we can write a = a^p, then if we can write a^-1 = (a^p)^-1, then we'll be good. But how the heck do I do that?

For closure under addition, I'm not really sure how to use the binomial theorem here, since we know that a + b means we can write them both as a^p + b^p, and now we've got to show that a^p + b^p can be written as (α + β)^p. Not sure how to get this one started either.

Any help is appreciated, thanks in advance.

 

[Dick]

Homework Statement

Homework Equations

The Attempt at a Solution

Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a^-1 is also in G.

So a*a^-1 = 1_F, but is a^-1 in G...so we can write a = a^p, then if we can write a^-1 = (a^p)^-1, then we'll be good. But how the heck do I do that?

For closure under addition, I'm not really sure how to use the binomial theorem here, since we know that a + b means we can write them both as a^p + b^p, and now we've got to show that a^p + b^p can be written as (α + β)^p. Not sure how to get this one started either.

Any help is appreciated, thanks in advance.

I'm not sure why you are stumbling around with the multiplicative proof. If a=a^p then a^(-1)=(a^p)^(-1). But that's the same thing as (a^(-1))^p isn't it? And for addition, you use the binomial theorem to prove that (a+b)^p=a^p+b^p in F. This is where the field having characteristic p is really important.