Finite Fields - (AKA Galois Fields) - Loyola University: Loyola

20 Pages·2008·0.09 MB·English

Finite Fields
(AKA Galois Fields)
November 24, 2008
() FiniteFields November24,2008 1/20
The Field of p Elements (Review)
By considering congruence mod n for any positive integers n we
constructed the ring Z = {0,1,2,...,n−1} of residue classes
n
mod n.
In Z we add, subtract, and multiply as usual in Z, with the
n
understanding that all multiples of n are declared to be zero in Z .
n
Algebraists often write Z = Z/nZ to emphasize the point that nZ,
n
the set of all multiples of n, is zero in the quotient Z = Z/nZ.
n
(Technically, nZ is an ideal in the ring Z and Z/nZ is the quotient
ring by that ideal.)
Using the Euclidean algorithm (which is based on the division
algorithm) one shows that a residue class a ∈ Z is invertible if
n
and only if gcd(a,n) = 1.
Hence, Z is a field1 if and only if n = p is a prime number.
n
1Inafieldeverynonzeroelementisinvertible.
() FiniteFields November24,2008 2/20
The Field of p Elements (Review)
Alternative notations for the field Z of p elements, when p is a
p
prime, are: F or GF(p) (GF stands for “Galois field.”).
p
Let’s use the F notation for Z henceforth, to emphasize the fact
p p
that we are dealing with a field and not just a ring.
GENERALIZATION
It turns out that there is a finite field F of q = pr elements, for
q
every prime power pr.
Moreover, there are no other examples of finite fields.
We will now discuss how to construct the finite fields F for q = pr
q
where r > 1.
() FiniteFields November24,2008 3/20
What is a ring, anyway?
A ring is a set R endowed with two operations called addition and
multiplication,suchthatthefollowingaxiomsholdforeverya,b,c ∈ R:
Addition is associative: a+(b+c) = (a+b)+c.
Addition is commutative: a+b = b+a.
Zero is neutral for addition: a+0 = a.
a has an opposite −a (in R) such that a+(−a) = 0.
Multiplication is associative: a(bc) = (ab)c.
The element 1 is neutral for multiplication: 1a = a = a1.
Multiplication distributes across addition: a(b+c) = ab+ac and
(a+b)c = ac+bc.
A commutative ring is a ring which also satisfies the law: ab = ba for
all a,b ∈ R.
() FiniteFields November24,2008 4/20
Fields, Ideals, and Quotient Rings
By definition, a field is just a commutative ring in which every
nonzero element has an inverse.
An ideal in a ring R is a nonempty subset J of R satisfying:
(a) a−b ∈ J for all a,b ∈ J (closure under subtraction);
(b) ra and ar are in J, for all a ∈ J, r ∈ R (closure under outside
multiplication).
Given an ideal J in a ring R, we define congruence mod J of two
given elements a,b ∈ R by declaring that a ≡ b (mod J) whenever
a−b ∈ J. This gives an equivalence relation on R. The resulting
equivalence classes are the elements of the quotient ring R/J.
For example, in R = Z the subset nZ is an ideal, and the resulting
quotient ring is Z = Z/nZ.
n
THEOREM: If R is a commutative ring and J an ideal then R/J is
a field if and only if J is a maximal ideal.
() FiniteFields November24,2008 5/20
The polynomial ring F [x]
p
The polynomial ring F [x] is the set of all polynomials with
p
coefficients from F . These are expressions of the form
p
f(x) = a +a x +a x2+···+a xn
0 1 2 n
where each coefficient a ∈ F . The set F [x] is an infinite set.
i p p
Recall that thedegree of a polynomial is the highest exponent of x
which occurs in the polynomial. For instance, the polynomial
x2+x +1 has degree 2 and 4x +1 has degree 1.
The degree of a constant polynomial is 0, except that by special
agreement we define the degree of the zero polynomial to be −1
or sometimes −∞.
() FiniteFields November24,2008 6/20
The polynomial ring F [x]
p
In any given degree, there are only finitely many elements of
F [x], since there are only a finite number of possibilites for each
p
coefficient.
We may add, subtract, and multiply polynomials in F as we
p
learned in school. Even though the coefficients are elements of F
p
instead of actual integers, it is easy to do the calculations so long
as we remember to always reduce coefficients mod p.
EXAMPLES. In F [x] we have:
5
(2x2+4x +1)+(4x3+3x2+4) = 4x3+4x;
(2x2+4x +1)−(4x3+3x2+4) = x3+4x2+4x +2;
(2x2+4x +1)·(4x3+3x2+4) = 3x5+2x4+x3+x2+x +4.
Check these calculations for yourself.
It is easy to check that F [x] is a commutative ring, for any prime
p
p.
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Division algorithm in F [x]
p
Of utmost importance is the fact that there is a division algorithm
in F [x] by which one can divide any given polynomial a(x) by
p
another (nonzero) polynomial b(x) and get a quotient polynomial
q(x) and a remainder polynomial r(x):
a(x) = b(x)q(x)+r(x), degr(x) < degb(x).
The method is the familiar long division of polynomials from
high-school algebra, except that one uses modular arithmetic on
the coefficients.
q(x)isthequotientpolynomialandr(x)theremainderpolynomial.
EXAMPLE. Check that
4x3+3x2+4 = (3x2+4x +1)(3x +2)+(4x +2), by dividing
a(x) = 4x3+3x2+4 by b(x) = 3x2+4x +1.
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Euclidean Algorithm for F [x]
p
The Euclidean algorithm may be extended to F [x]. It works just
p
the same in F [x] as it does in Z, except that one must replace
p
ordinary long division of integers by long division of polynomials in
F [x].
p
The extended Euclidean algorithm also works: by working
backwards with the equations coming from the Euclidean
algorithm,onecanalwaysfindpolynomialss(x)andt(x)suchthat
gcd(a(x),b(x)) = a(x)s(x)+b(x)t(x).
Any commutative ring without zero divisors in which the Euclidean
algorithm holds is called a Euclidean domain. The ring of
polynomials with coefficients in a field is always a Euclidean
domain.
() FiniteFields November24,2008 9/20
Divisibility in F [x]
p
Let a(x),b(x) ∈ F [x]. Say that a(x) divides b(x) (written as
p
a(x) | b(x)) if there is a polynomial q(x) ∈ F [x] such that
p
b(x) = a(x)q(x).
Every polynomial has lots of trivial factorizations. Since any
nonzero α ∈ F is invertible, we can always write
p
a(x) = α·α−1a(x) where α−1a(x) ∈ F [x]. But note that in such a
p
trivial factorization, the degree of the factors is 0 and dega(x).
We define a nontrivial factorization of a(x) to be one of the form
a(x) = b(x)c(x) where the degrees of b(x),c(x) are both strictly
positive.
Definition: A polynomial a(x) is said to be irreducible in F [x] if it
p
has no factorizations other than the trivial ones. For example,
x2+x +1 is irreducible in F [x], as you should check.
2
() FiniteFields November24,2008 10/20