Prime Numbers

92 Chapter 2 NUMBER-THEORETICAL TOOLS
h(x) ∈ F [x] if and only if G(x) ≡ g(x) (modf(x)). Thus, working with cosets
can be thought of as a fancy way of working with congruences.) Each coset
has a canonical representative, that is, a unique and natural choice, which is
either 0 or has degree smaller than deg f(x).
We can add and multiply cosets by doing the same with their representatives:
g1(x)+(f(x)) + g2(x)+(f(x)) = g1(x)+g2(x) + (f(x)),
g1(x)+(f(x)) · g2(x)+(f(x)) = g1(x)g2(x) + (f(x)).
With these rules for addition and multiplication, F [x]/(f(x)) is a ring that
contains an isomorphic copy of the field F :Anelementa ∈ F is identified
with the coset a +(f(x)).
Theorem 2.2.3. If F isafieldandf(x) ∈ F [x] has positive degree, then
F [x]/(f(x)) isafieldifandonlyiff(x) is irreducible in F [x].
Via this theorem we can create new fields out of old fields. For example,
starting with Q, the field of rational numbers, consider the irreducible
polynomial x 2 − 2inQ[x]. Let us denote the coset a + bx +(f(x)), where
a, b ∈ Q, moresimplybya + bx. We have the addition and multiplication
rules
(a1 + b1x)+(a2 + b2x) =(a1 + a2)+(b1 + b2)x,
(a1 + b1x) · (a2 + b2x) =(a1a2 +2b1b2)+(a1b2 + a2b1)x.
That is, one performs ordinary addition and multiplication of polynomials,
except that the relation x2 = 2 is used for reduction. We have “created” the
field
√
Q 2 = a + b √
2:a, b ∈ Q .
Let us try this idea starting from the finite field F7. Saywetakef(x) =
x 2 + 1. A degree-2 polynomial is irreducible over a field F ifandonlyifithas
no roots in F . A quick check shows that x 2 + 1 has no roots in F7, soitis
irreducible over this field. Thus, by Theorem 2.2.3, F7[x]/(x 2 +1)isafield.
We can abbreviate elements by a + bi, wherea, b ∈ F7 and i 2 = −1. Our new
field has 49 elements.
More generally, if p is prime and f(x) ∈ Fp[x] is irreducible and has
degree d ≥ 1, then Fp[x]/(f(x)) is again a finite field, and it has p d elements.
Interestingly, all finite fields up to isomorphism can be constructed in this
manner.
An important difference between finite fields and fields such as Q and C
is that repeatedly adding 1 to itself in a finite field, you will eventually get 0.
In fact, the number of times must be a prime, for otherwise, one can get the
product of two nonzero elements being 0.
Definition 2.2.4. The characteristic of a field is the additive order of 1,
unless said order is infinite, in which case the characteristic is 0.