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).
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.
I think that it's perhaps because I made an EGREGIOUS typo/mistake in my earlier post (what time of night was that)? Ok, so you're trying to show that but since is an isomorphism this is EQUIVALENT to showing that (we know that since is the identity of and so cooresponds to the identity of ). But, since is a morphism we know that from there I think you got the reason why this is zero.
I'm sorry for the error, does that make more sense?
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)?