1.13.2 The Field ℤp
With this preparation, here is the construction of the finite fields ℤp for p a prime.
Definition 1.13.8 Let ℤ+ denote the set of nonnegative integers. Thus ℤ+ =
let p be a prime number. We will say that two integers, a,b are equivalent and write a ∼ b if a − b
is divisible by p. Thus they are equivalent if a − b
= px for some integer x.
Proposition 1.13.9 The relation ∼ is an equivalence relation. Denoting by n the equivalence class
determined by n ∈ ℕ, the following are well defined operations.
which makes the set ℤp consisting of
into a field.
Proof: First note that for n ∈ ℤ+ there always exists r ∈
. This is
clearly true because if n ∈ ℤ+
, then n
for r < p
, this by the Euclidean algorithm. Thus
. Now suppose that n1
Is it true that n1 + m1
= n + m
? Is it true that
is a multiple of
? Of course since n1 − n
and m1 − m
are both multiples
. Similarly, is n1m1
? Is nm − n1m1
a multiple of p
? Of course this is so because
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 ℤp
is of the form r
0 ≤ r ≤ p −
By definition, r
+ p − r
and so the additive inverse
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
Choose m ≥ 0 such that pm + x > 0,pm + y > 0. Then
It follows that 1 + pmn + p2m = 1
is the multiplicative inverse of n. ■
Thus ℤp 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 1.13.10 A commutative ring with unity is just a field except it lacks the property that
nonzero elements have a multiplicative inverse. It has all other properties. Thus the axioms of a
commutative ring with unity are as follows:
Axiom 1.13.11 Here 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 + = 0
, (existence of additive inverse).
z = x +
,(associative law for addition).
- xy = yx,(commutative law for multiplication). You could write this as x × y = y × x.
z = x
,(associative law for multiplication).
- There exists 1 such that 1x = x for all x,(multiplicative identity).
- x =
xy + xz.(distributive law).
An example of such a thing is ℤm where m is not prime, also the ordinary integers. However, the
integers are also an integral domain.
Definition 1.13.12 A commutative ring with unity is called an integral domain if, in addition to
the above, whenever ab = 0, it follows that either a = 0 or b = 0.