Page 240 - DMTH403_ABSTRACT_ALGEBRA

Unit 24: Irreducibility and Field Extensions

Thus, the prime fields are Q, Z , Z , Z , etc. Notes
2 3 5
We call the subfield isomorphic to a prime field (obtained in Theorem 6), the prime subfield of
the given field.
Let us again reword Theorem 6 in terms of field extensions. What it says is that every field is a
Weld extension of a prime field.
Now, suppose a field F is an extension of a field K. Are the prime subfields of K and F isomorphic
or not? To answer this let us look at char K and char F. We want to know if char K = char F or not.
Since F is a field extension of K, the unity of F and K is the same, namely, 1. Therefore, the least
positive integer n such that n.1 = 0 is the same for F as well as K. Thus, char K = char F. Therefore,
the prime subfields of K and F are isomorphic.
A very important fact that a field is a prime field iff it has no proper subfields.
Now let us look at certain field extensions of the fields Z .
p
24.2.2 Finite Fields

You have dealt a lot with the finite fields Z . Now we will look at field extensions of these fields.
p
You know that any finite field F has characteristic p, for some prime p. And then F is an extension
of Z. Suppose P contains q elements. Then q must be a power of p. That is what we will prove
now.
Theorem 7: Let F be a finite field having q elements and characteristic p. Then q = p , some
n
positive integer n.
The proof of this result uses the concepts of a vector space and its basis.
Proof: Since char F = p, F has a prime subfield which is isomorphic to Z . We lose nothing if we
p
assume that the prime subfield is Z . We first show that F is a vector space over Z with finite
p
p
dimension.
Recall that a set V is a vector space over a field K if:
(i) we can define a binary operation + on V such that (V, +) is an abelian group,
(ii) we can define a ‘scalar multiplication. : K × V  V such that  a, b  K and v, w  V,
a(a + w) = a.v + a.w

(a + b).v = a.v + b.w
(ab). v = a. (b.v)
1.v = v.

Now, we know that (F, +) is an abelian group. We also know that the multiplication in F will
satisfy all the conditions that the scalar multiplication should satisfy. Thus, F is a vector space
over 2,. Since F is a finite field, it has a finite dimension over Z . Let dim Z F = n. Then we can
p
p
find a,. .., a , a F such that
n
F = Z a + Z a + .. + Z a .
p 2
p 1
p n
We will show that F has pn elements.
Now, any element of F is of the form
b a , + b a + ... +, b a , where b,, . .., b  Zp.
n n
2 2
n
1 1
Now, since o(Zp) = p, b can be any one of its p elements.
1

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