Math 155: Designs and groups
Handout #2 (1 February 2010): Finite fields
Since we’re studying finite combinatorial structures, we’ll have to do algebra and
linear algebra over finite fields. The most familiar of these are the prime fields
Fp=Z/pZwhere p∈Zis a prime. In general any finite field Fcontains a unique
prime field, consisting of all the elements of Fof the form 1 + 1 + . . . + 1. The size,
call it again p, of this prime field is the characteristic of F. Since Fis a vector space
over Fpwe have #F=pnfor some natural number n(namely the dimension of
that vector space). We cite without proof the following fundamental theorem, due
in essence to Galois:
For each prime pand integer n>1 there exists a finite field F
of cardinality pn. This field is unique up to isomorphism. The
automorphism group of Fis canonically isomorphic with Z/nZ
and is generated by the Frobenius automorphism x7→ xp. For each
positive divisor mof n, that field contains a unique subfield F1of
cardinality qm, namely {x:xqm=x}. The field extension F/F1
is normal, with cyclic Galois group of order m/n generated by
x7→ xpm.
We shall use F
qfor the finite field of cardinality q=pn; the older notation GF(q)
for F
q(“GF” as in “Galois field”) is still occasionally seen in the literature. These
fields are a natural and important generalization of the familiar prime fields Fp;
in general anything that can be done with Fpworks just as well with Fq, and
sometimes one can do a bit more with the non-prime fields thanks to the nontrivial
automorphisms (as is true for C, which though less familiar than Rturns out to
be equally fundamental and sometimes more tractable). For example, you probably
know that for every prime pthere is at least one “primitive residue” mod p, which
is to say that the multiplicative group F∗
pis cyclic; the same is true (with much the
same proof) for F∗
qfor any finite field Fq. Warning: once n > 1, the finite field of
pnelements is not Z/pnZ(and its additive group is not cyclic).
Except for the familiar prime fields (with n=1), the only finite fields we shall have
much use for are F
4and F9; these may be defined as the quadratic extensions F2(ρ)
and F3(i) of their prime fields, where ρ2+ρ= 1 and iis a square root of −1.
