Sophie Germain: mathématicienne extraordinaire - Scripps College

The First Proof 43
us
5 3 r 4 + 2 · 5 2 r 2 q 2 + q 4 = (q 2 + 5 2 r 2 ) 2 − (5 4 r 4 ) − (5 3 r 4 )
= (q 2 + 5 2 r 2 ) 2 − 5(10r 2 ) 2
Let P 0 = (q 2 + 5 2 r 2 ) and let Q 0 = 10r 2 so that P 0 and Q 0 are both
positive integers such that
(q 2 + 5 2 r 2 ) 2 − 5(10r 2 ) 2 = P 2 0 − 5Q 2 0
= (P 0 + Q 0
√
5)(P0 − Q 0
√
5).
Now for something unexpected and exciting: we are leaving the integers.
We are in fact entering the ring of integers of the number field
Q( √ 5). Recall that from the Fundamental Theorem of Arithmetic, we have
unique factorization in the integers. That is, every integer m ≠ 0 can be
factored into a unique product of primes up to a unit. In the integers, our
units are ±1. Thus we say that every integer m ≥ 1 can be factored into a
unique product of positive primes. Similarly, we say that unique factorization
holds in a number field if every element can be factored into a unique
product of irreducible elements up to a unit. Naturally, we ask ourselves,
what does an element of a number field look like and how does it factor
Definition 1. The number field Q( √ p) where p is square free, is the set of elements
a + b √ p such that a and b are rational numbers.
In particular, we are interested in Q( √ 5) = {a + b √ 5 : a, b ∈ Q}.
We
can similarly define the ring Z[ √ 5] = {a + b √ 5 : a, b ∈ Z}. We would like