文档介绍：page 1 of Chapter 7
CHAPTER 7 INTRODUCING ALGEBRAIC NUMBER THEORY
(COMMUTATIVE ALGEBRA 1)
The general theory mutative rings is known mutativealgebra . The
main applications of this discipline are to algebraic number theory, to be discussed in this
chapter, and algebraic geometry, to be introduced in Chapter 8.
Techniques of abstract algebra have been applied to problems in number theory for
a long time, notably in the effort to prove Fermat’s Last Theorem. As an introductory
example, we will sketch a problem for which an algebraic approach works very well. If p is
an odd prime and p ≡ 1 mod 4, we will prove that p is the sum of two squares, that is, p
2 2 p−1
can be expressed as x + y where x and y are integers. Since 2 is even, it follows that
-1 is a quadratic residue (that is, a square) mod p. [Pair each of the numbers 2,3,...,p − 2
with its inverse mod p and pair 1 with p − 1 ≡−1modp. The product of the numbers 1
through p − 1 is, mod p,
p − 1 p − 1
1 × 2 ×···× ×−1 ×−2 ×···×−
2 2
p−1 2 ≡−
and therefore [( 2 )!] 1modp.]
If −1 ≡ x2 mod p, then p divides x2 + 1. Now we enter the ring of Gaussian integers
and factor x2 + 1 as (x + i)(x − i). Since p can divide neither factor, it follows that p is not
prime in Z[i], so we can write p = αβ where neither α nor β is a unit.
Define the norm of γ= a + bi as N(γ)=a2 + b2. Then N(γ)=1iffγ= ±1or±i iff
γ is a unit. (See Section , Problem 5.) Thus
p2 = N(p)=N(α)N(β) with N(α) > 1 and N(β) > 1,
so N(α)=N(β)== x + iy, then p = x2 + y2.
Conversely, if p is an odd prime and p = x2 + y2, then p is congruent to 1 mod 4. (If x
is even, then x2 ≡ 0 mod 4, and if x is odd, then x2 ≡ 1 mod 4. We cannot have x and y
both even or both odd, since p is odd.)
It is natural to conjecture that we can identify those primes that√ can be represented as
x2 + |d|y2, where d is a negative integer, by working in the ring Z[ d]. But the Gaussian
integers (d = −1) form a Euclidean domain, in particular a