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LECTURE 7: CLASSIFICATION OF FINITE FIELDS
To begin with we show that every finite field must have pn elements for some
prime p and n P N.
Theorem. Let E : F be a finite extension of a finite field F . If rE : F s “ n and
#F “ q, then #E “ q n .
Proof. The key ideas of the proof are:
‚ Let tα1 , . . . , αn u be a basis for E over F . Each β P E can be uniquely
written as
n
ÿ
β“
bj αj
j“1
for some choice of bj P F for 1 ď j ď n.
‚ There are q n choices of coefficients bj , each giving a distinct element of E.
Corollary. If F is a finite field and charF “ p, then #F “ pn for some n P N.
We can actually determine precisely what a field of cardinality pn should look
like, if it exists.
Theorem. Let Z̄p be an algebraic closure of Zp and E Ď Z̄p be a field with #E “
pn . Then
n
E “ tx P Z̄p : xp ´ x “ 0u.
(1)
Proof. The key ideas of the proof are:
‚ The multiplicative group E ˚ :“ Ezt0u has order pn ´ 1.
n
‚ By Lagrange, ordpαq|pn ´ 1 for each α P E ˚ and so αp ´1 ´ 1 “ 0. This
pn
shows each element of E is a root of x ´ x.
n
‚ Since xp ´ x can have at most pn roots, E is precisely the set of roots.
We have shown that any finite field must be of the form described in (1). To
classify all finite fields, it suffices to show that each set of the form (1) does indeed
define a finite field and has cardinality pn .
Theorem. For each p prime and n P N the set
n
E :“ tx P Z̄p : xp ´ x “ 0u
is a subfield of Z̄p of cardinality pn .
Proof. The key ideas of the proof are:
‚ E is clearly closed under multiplication.
n
n
n
‚ For α, β P Z̄p , prove pα ` βqp “ αp ` β p by induction. This shows E is
closed under addition.
‚ It remains to show #E “ pn . Thus would follow if every zero f pxq :“
n
xp ´1 ´ 1 is of multiplicity 1.
‚ Suppose α is a zero of f pxq so
f pxq “ px ´ αqgpxq :“ px ´ αqpxp
n
´2
` αxp
n
´3
` ¨ ¨ ¨ ` αp
n
´3
‚ Note α is a zero of multiplicity 1 if and only if gpαq ‰ 0.
1
x ` αp
n
´2
q.
2
LECTURE 7: CLASSIFICATION OF FINITE FIELDS
‚ Since
gpαq “ ppn ´ 1qαp
n
´2
“ ppn ´ 1qα´1 “ ´α´1 ‰ 0,
each zero of f has multiplicity 1.
Theorem. If E1 , E2 are finite fields and #E1 “ #E2 , then E1 – E2 .
The key point here is that although E1 and E2 may lie in different algebraic
closures of Zp , they are nevertheless isomorphic as fields.
To prove the theorem we using the following lemma.
Lemma. Any finite extension E : Zp is a simple algebraic extension.
Proof. The key ideas of the proof are:
‚ E ˚ is cyclic (proved in an earlier quarter) and so E “ Zp pαq where α is a
generator of E ˚ .
Proof (of Theorem). The key ideas of the proof are:
‚ #E1 “ #E2 “ pn for some prime p and n P N. Thus they both are finite
extensions of the prime field Zp .
‚ Let E1 “ Zp pαq so E1 – Zp rxs{xf y where f “ irrpα, Zp q.
n
‚ Since every element of E1 is a root of xp ´ x, it follows that f is a factor
n
xp ´ x.
n
‚ E2 contains all the roots of xp ´ x and hence all the roots of f .
‚ E2 must contain a subfield which is isomorphic to Zp rxs{xf y.
‚ Since #E2 “ pn , we conclude that E2 – Zp rxs{xf y.
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]