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## Algebraic Number Theory (Paul Tartell, Mike Pepper, David Grant, 1976)

(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.