(to the tune of Both Sides Now)
Rings and fields and U.F.D.
Educlidean Algorithm for me
We've seen it done at first in Z
And looked at primes that way.
But George and Eric are having fun
They lead us on and we seem dumb
Subtilties aren't the only ones
That get into our way.
I've looked at primes from both sides now
From norms and forms and still somehow
It's prime confusions I recall
I really don't know primes at all.
x^2 + 2y^2's primes
Have been known to us for some time
1 (mod 8) and also 3
and 2 thrown in for free
But Fermat and Pell are straightening out
The thing that first engendered doubt
Problems that are diophantine
Have all become routine.
I've looked at these equations from both sides now,
From trial to error and still somehow
It's algebraics I recall
For giving these forms and overhaul.
Proofs of Quadratic Reciprocity
I love the roots of unity
The congruences qua Z mod p
And some trigonometry.
Gaussian sums are truly neat
They make our prooofs really quite complete
A on P and P on A
Turn up in many ways.
I've looked at residues from both sides now
From Legendre to Jacobi, and still somehow
It's Eisenstein's proof that I recall
Mixed in with Robert Klein's twilight call.