Basic concepts: Field extensions and irreducible polynomials, construction of finite fields, primitive elements.
We have seen in the preceding section thatZ/pZ=Fpis a finite field when pis prime. Are there any other finite fields? It is easy to see the following: if nis the number of elements (the order) of a fieldK,thenn=prmust be a power of a prime. Also, each field of orderprmust contain the prime fieldFp. The prime pis then called the characteristic of K.We do not want to go too deeply into field theory. The following theorem will be accepted without proof:
3.6 Theorem. For every prime-powerprthere is exactly one fieldFpr of that order.
We need to know how to construct these fields. AsFpr contains the prime field Fp, it is anr-dimensional vector space overFp. We can think ofFpr as the space Frp of r-tuples with entries in the prime field, with an additional multiplicative structure.
In this section we want to describe how finite fields are constructed and use as an example the smallest case F4.The first (and in general most difficult) step in the construction of Fpr is to find a polynomialXr+. . . of degreer with coefficients inFp,which isirreducible.
In our example we simply write down all binary polynomials of degreer= 2.
Fortunately, there are only four such polynomials:
X2, X2+ 1, X2+X andX2+X+ 1.
A polynomial of degreeris irreducible if it cannot be written as the product of two polynomials of smaller degree. Irreducible polynomials play a role, which is analogous to the primes in the context of integers. In our example we see that X2 =X·X, X2+ 1 = (X+ 1)2 and X2+X =X(X+ 1) are certainly not irreducible. The only remaining candidate isX2+X+ 1.This is an irreducible polynomial.
Assume now we have found an irreducible polynomial f(X) =Xr+ar−1Xr−1+· · ·+a1X+a0.
The elements ofFpr are the polynomials of degree < r.Addition and multi- plication are as usual with polynomials. The important simplification is that f(X) = 0. We say we calculatemod the polynomialf(X).When doing this it is customary to assign a neutral Greek name to the (former) variable X, sayǫ.The defining equation is thereforef(ǫ) = 0.This means that, whenever a power ofǫappears with an exponent≥r, we reduce the exponent by using the defining equation
ǫr=−ar−1ǫr−1− · · · −a1ǫ−a0.
Back to the exampleF4.We know thatf(X) =X2+X+ 1 is our irreducible polynomial (with coefficients inF2). As a new name for X when calculating modf(X) we chooseω.The defining equation is therefore
ω2=ω+ 1
(observe that in characteristic 2 we have + = −). This equation describes the field structure completely: F4={0,1, ω, ω+ 1} and addition is addition of polynomials. The defining equation tells as what ω2 is. We can go on:
ω3 = ω(ω+ 1) = ω2+ω = (ω + 1) +ω = 1. As ω3 = 1, we say that ω hasorder3.We have written all nonzero field elements as powers ofω.This determines the multiplicative structure.
In general it is not always true that all field elements are powers ofX.If it is the case, one speaks of aprimitivepolynomial and calls the corresponding
field elementsprimitive elements. Every finite field can be described by a primitive polynomial.
We want to accept that fields Fq exist (and are uniquely determined) for every prime-powerq.It will be assumed that the reader is completely familiar with prime fields and with F4. Should we encounter any other finite field, it will be constructed following the recipe given in this section.
Although we need it here only over finite fields, the same procedure shows how to construct field extensions in general:
3.7 Theorem. LetK be a field andp(X)an irreducible polynomial of degree rwith coefficients inK.Then K[X]/(p(X))is a field extension of degreerof K.
Here ((p(X)) denotes the set of all polynomials that are multiples ofp(X) and the factor notationK[X]/(p(X)) is a shorthand for the construction we saw: the powers 1, X, . . . , Xr−1 are a basis of this extension field overK. In Exercise 3.2.5 the reader is asked to construct the complex number fieldCas an extension of the realsRin this way.
1. An irreducible polynomial of degree r with coefficients in field Kyields a degreerextension field ofK.
2. For every prime-powerq=prthere is a finite field Fq withqelements.
3. It is uniquely determined and contains the prime fieldFp. 4. Fpr is constructed with the help of an irreducible
polynomial of degreer with coefficients inFp. 5. The fieldF4 has been described in detail.
6. F4={0,1, ω, ω2},where 1 +ω+ω2= 0 and ω3= 1.
Exercises 3.2
3.2.1. Find all irreducible polynomials of degree3 with coefficients in F2. 3.2.2. ConstructF8.
3.2.3. Is it true thatF4⊂F8?
3.2.4. Determine the sum of all elements inFq.
3.2.5. Construct the field of complex numbers as a quadratic extension of the fieldR of real numbers. What is the natural choice of a quadratic irreducible polynomial?
3.2.6. ConstructF9,using the irreducible polynomial X2−X−1.
3.2.7. Let α∈F =Fqr. Show that there is a polynomialf(X)of degree≤r with coefficients inFq such that f(α) = 0.
3.2.8. Let α∈F =Fqr. Show that there is a unique monic (highest coeffi- cient= 1) polynomial f(X)of smallest degree s >0 such thatf(α) = 0.The polynomialf(X)is the minimal polynomialof α.It is irreducible.
3.2.9. Let α∈F =Fqr.Let sbe the degree of the minimal polynomial ofα, equivalently the smallest number such that the powers1, α, . . . , αsare linearly dependent over Fq. Show that the smallest subfield Fq(α)of F containingFq andαhas 1, α, . . . , αs−1 as basis. Conclude thats must divide r.
3.2.10. In the following exercises readers are encouraged to prove some of the basic facts concerning finite fields. Start with the following: let F be a finite field. For every natural numbern definen·1 = 1 + 1 +· · ·+ 1as the sum of ncopies of 1 in F.
Prove that there must exist some n >0 such thatn·1 = 0.
3.2.11. LetF be a finite field. Define thecharacteristicofF as the smallest natural numberpsuch thatp·1 = 0.
Prove that the characteristic is a prime number.
3.2.12. Let p be the characteristic of the finite field F. Prove that {0,1,2· 1, . . . ,(p−1)·1}is a subfield ofF and that this subfield is the prime fieldFp. 3.2.13. Let F be a finite field of characteristic p. Prove that F is a vector space overFp. Conclude: |F|must be a power of p.