Question on proof regarding structure of finite fields

I am reading the proof of the fact that as a group under multiplication, the set of nonzero elements of (the Galois field/finite field of order ) is isomorphic to and is therefore cyclic.

During the course of the proof, the author reminds the reader that the operation is componentwise addition. I do not see why this is true...we are looking at the multiplicative group of the field. Can someone explain? (I will add more detail if necessary).

Re: Question on proof regarding structure of finite fields

Quote:

Originally Posted by

**AlexP** I am reading the proof of the fact that as a group under multiplication, the set of nonzero elements of

(the Galois field/finite field of order

) is isomorphic to

and is therefore cyclic.

During the course of the proof, the author reminds the reader that the operation is componentwise addition. I do not see why this is true...we are looking at the multiplicative group of the field. Can someone explain? (I will add more detail if necessary).

This doesn't make sense, we need more context. What would he even mean by component wise addition? Perhaps thinking of in its usual form as for an appropriate from where the 'addition componentwise' would make sense for the addition in .

Re: Question on proof regarding structure of finite fields

True...the componentwise part doesn't make sense without the rest. Here's the whole proof. There's another part I don't understand so I'll have that part ready too."

"To see that the multiplicative group of nonzero elements of is cyclic, we first note...that it is isomorphic to a direct product of the form , where each divides . So, for any element in this product, we have . (Remember, the operation is componentwise addition.)

Thus, the polynomial has zeros in . Since the number of zeros of a polynomial over a field cannot exceed the degree of the polynomial, we know that . On the other hand, since has a subgroup isomorphic to , we also have . It follows, then, that is isomorphic to ."

Re: Question on proof regarding structure of finite fields

Quote:

Originally Posted by

**AlexP** True...the componentwise part doesn't make sense without the rest. Here's the whole proof. There's another part I don't understand so I'll have that part ready too."

"To see that the multiplicative group

of nonzero elements of

is cyclic, we first note...that it is isomorphic to a direct product of the form

, where each

divides

. So, for any element

in this product, we have

. (Remember, the operation is componentwise addition.)

Thus, the polynomial

has

zeros in

. Since the number of zeros of a polynomial over a field cannot exceed the degree of the polynomial, we know that

. On the other hand, since

has a subgroup isomorphic to

, we also have

. It follows, then, that

is isomorphic to

."

First of all, don't use tensor for direct product since that has a separate meaning. What he is really saying is this, if is an isomorphism we have that where I made use of the fact that since, if we have that since for each and so by Lagrange's theorem. Make sense?

Also if you are having trouble understanding this I, at the risk of shamelessly self-promoting, suggest you see this post I wrote on my blog about this topic.

Re: Question on proof regarding structure of finite fields

I understand that , but with I'm getting which does not make sense.

That notation was a mistake, no idea what I was thinking. I meant to use what the book uses, which is .

Re: Question on proof regarding structure of finite fields

more generally, any finite subgroup of the multiplicative group of a field is cyclic.

this result, under some conditions, can be extended nicely to division rings. see here.

Re: Question on proof regarding structure of finite fields

Quote:

Originally Posted by

**AlexP** I understand that

, but with

I'm getting

which does not make sense.

That notation was a mistake, no idea what I was thinking. I meant to use what the book uses, which is

.

How exactly do you manage that? Perhaps if you told us how you are getting that result we can help find your mistake.

Re: Question on proof regarding structure of finite fields

Wow...I have no idea where my brain has been lately. That's scary.

What you *did* say was , and that's where I got that. I'm clearly not following something.

Re: Question on proof regarding structure of finite fields

Re: Question on proof regarding structure of finite fields

It's fine, everyone's entitled to some mistakes (I certainly made some as well). I do see it clearly now. Just one thing... You wrote our map as , but should it really be (specifically from the multiplicative group of our field to the direct product)?

Re: Question on proof regarding structure of finite fields

Quote:

Originally Posted by

**AlexP** It's fine, everyone's entitled to some mistakes (I certainly made some as well). I do see it clearly now. Just one thing... You wrote our map as

, but should it really be

(specifically from the multiplicative group of our field to the direct product)?

Yes.

Re: Question on proof regarding structure of finite fields

ok. Thank you very much for the help (and patience).

Re: Question on proof regarding structure of finite fields

Quote:

Originally Posted by

**AlexP** ok. Thank you very much for the help (and patience).

Anytime :)