Using group theory to prove results in number theory

I was interested in the subject of quadratic residues recently. Let be an odd prime. We say that an integer is a quadratic residue iff there exists an integer such that otherwise we say that is a quadratic non-residue

Let be the set of all quadratic residues Then, you know what? is a subgroup of index of the multiplicative group of the field of integers This is not mentioned in a number of books and websites on quadratic residues I've seen, but I think a development based on this group-theoretic approach can advance our understanding of results in number theory, in particular quadratic residues.

Watch the following space. (Wink)

Re: Using group theory to prove results in number theory

Let be an odd prime and the set of all quadratic residues

__Lemma 1:__

__Proof:__

We have But note that Hence we can write since This shows that there are at most distinct quadratic residues But now suppose that and Then or The latter would imply that which is impossible for Hence if This proves that there are exactly quadratic residues

Re: Using group theory to prove results in number theory

Re: Using group theory to prove results in number theory

Re: Using group theory to prove results in number theory

Re: Using group theory to prove results in number theory

Re: Using group theory to prove results in number theory

Quote:

Originally Posted by

**Sylvia104** We prove the next result, which is Euler's criterion:

__T2:__ If

is an odd prime and

is an integer coprime with

then

There are two things to note before we proceed with the proof. First,

can only be

This is because

by Fermat

and Fermat by Lagrange. indeed Fermat's "little theorem" generalizes quite naturally to one of Euler: , if

Quote:

and

Second, if

is a quadratic non-residue

then

can be expressed as

where

is primitive root

and

This again comes from the coset property of

more about this later. (Wink)

i am looking forward to your proof that the "other coset" of must necessarily contain a primitive root (i suspect you'll need to show we are dealing with cyclic groups).