With this preparation, here is the construction of the finite fields ℤ_{p} for p a prime.
Definition 1.13.8Let ℤ^{+}denote the set of nonnegative integers. Thus ℤ^{+} =
{0,1,2,3,⋅⋅⋅}
. Alsolet p be a prime number. We will say that two integers, a,b are equivalent and write a ∼ b if a − bis divisible by p. Thus they are equivalent if a − b = px for some integer x.
Proposition 1.13.9The relation ∼ is an equivalence relation. Denoting bynthe equivalence classdetermined by n ∈ ℕ, the following are well defined operations.
¯n+ m¯≡ n-+-m-
---
¯nm¯≡ nm
which makes the set ℤ_{p}consisting of
{ ----}
¯0,¯1,⋅⋅⋅,p− 1
into a field.
Proof: First note that for n ∈ ℤ^{+} there always exists r ∈
{0,1,⋅⋅⋅,p− 1}
such that n = r. This is
clearly true because if n ∈ ℤ^{+}, then n = mp + r for r < p, this by the Euclidean algorithm. Thus
r = n. Now suppose that n_{1} = n and m_{1} = m. Is it true that n_{1} + m_{1} =n + m? Is it true that
(n + m)
−
(n1 + m1)
is a multiple of p? Of course since n_{1}− n and m_{1}− m are both multiples
of p. Similarly, is n_{1}m_{1} =nm? Is nm − n_{1}m_{1} a multiple of p? Of course this is so because
nm − n m = nm − n m + n m − n m
1 1 1 1 1 1
= m (n − n1)+ n1(m − m1 )
which is a multiple of p. Thus the operations are well defined. It follows that all of the field
axioms hold except possibly the existence of a multiplicative inverse and an additive inverse.
First consider the question of an additive inverse. A typical thing in ℤ_{p} is of the form r where
0 ≤ r ≤ p − 1. Then consider
-----
(p − r)
. By definition, r +p − r = p = 0 and so the additive inverse
exists.
Now consider the existence of a multiplicative inverse. This is where p is prime is used. Say n≠0. That
is, n is not a multiple of p, 0 ≤ n < p. Then since p is prime, n,p are relatively prime and so there are
integers x,y such that
1 = xn+ yp
Choose m ≥ 0 such that pm + x > 0,pm + y > 0. Then
1 + pmn + pmp = (pm + x)n+ (pm + y)p
It follows that 1 + pmn + p^{2}m = 1
--------
¯1 = (pm + x)¯n
and so
(pm + x)
is the multiplicative inverse of n. ■
Thus ℤ_{p} is a finite field, known as the field of residue classes modulo p.
Something else which is often considered is a commutative ring with unity.
Definition 1.13.10A commutative ringwith unity is just a field except it lacks the property thatnonzero elements have a multiplicative inverse. It has all other properties. Thus the axioms of acommutative ring with unity are as follows:
Axiom 1.13.11Here are the axioms for a commutative ring with unity.
x + y = y + x, (commutative law for addition)
There exists 0 such that x + 0 = x for all x, (additive identity).
For each x ∈ F, there exists −x ∈ F such that x +
(− x)
= 0, (existence of additive inverse).
(x +y)
+ z = x +
(y+ z)
,(associative law for addition).
xy = yx,(commutative law for multiplication). You could write this as x × y = y × x.
(xy)
z = x
(yz)
,(associative law for multiplication).
There exists 1 such that 1x = x for all x,(multiplicative identity).
x
(y +z)
= xy + xz.(distributive law).
An example of such a thing is ℤ_{m} where m is not prime, also the ordinary integers. However, the
integers are also an integral domain.
Definition 1.13.12A commutative ring with unity is called anintegral domain if, in addition tothe above, whenever ab = 0, it follows that either a = 0 or b = 0.