(Should return to this later, but the clean conceptual route is of course to work with Hurwitz quaternions, the -span of where .)
The four squares theorem says that every natural number can be expressed as a sum of four integer squares. For this it is useful to recall that for quaternions , the central elements are real and are precisely the fixed points of the anti-involution . It follows quickly that the norm map is multiplicative: we have
for all quaternions .
This shows that sums of four integral squares are closed under multiplication, which reduces the four squares theorem to the statement that every prime is a sum of four squares. For we have .
Lemma
If a finite field has an odd number of elements , then every element of is a sum of two squares. (The conclusion holds also in characteristic , but we don’t need this.)
Proof
If some non-square is not a sum of any two squares, then this is true of every non-square (since must be a square, using cyclicity of the group ), in which case it follows that the squares are closed under addition and form a proper subfield , with elements. If , then , which is impossible.
Lemma
If is an odd prime, then there is some with such that is a sum of four squares.
Proof
We just saw that is a sum of two squares in , say where are represented in the range . Then divides . It follows that some with is a sum of four squares.
Lemma
If is a sum of four squares, then so is .
Proof
Writing , we have , so we may assume WLOG that have the same parity as then do . Then .
Theorem
Every odd prime is the sum of four squares.
Proof
Take to be the least element such that is of the form (invoking Lemma ). The claim is that ; since is prime, it suffices to show . By Lemma and minimality, is odd. Write , so . Take so that
Put . Then , and , so . Thus we may write in the form .
We have , and , so there is with such that . Thus , whence . This forces by minimality of . So , forcing , and now we must conclude . Thus divides , so divides and we are done.
Revised on June 8, 2020 at 16:25:54
by
Todd Trimble