Page 239 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Now, given any f(x)  F[x], such that deg f(x) > 0, we will show that there is a field monomorphism
from F into F[x]/d(x)>. This will show that F[x)/<f(x)> contains an isomorphic copy of F; and
hence, we can say, that it contains F. So, let us define 0 : F  F[x]/d(x)>: (a) = a + <f(x)>.
Then  (a+b) =  (a) +  (b), and

(ab) = (a) (b).
Thus,  is a ring homomorphism.
What is Ker  ?
Ker  = {a  F] a + <f(x)> = <f(x)>}

= {a  F | a  <f(x)>}
= {a  F | f(x) | a}
= {0}, since deg f > 0 and deg a  0.
Thus,  is 1-1, and hence an inclusion.

Hence, F is embedded in F[x]/<f(x)>.
Thus, if f(x) is irreducible in F[x], then F[x]/<f(x)> is a field extension of F.
Well, we have looked at field extensions of any field F. Now let us look at certain fields, one of
which F will be an extension of.

24.2.1 Prime Fields

Let us consider any field F. Can we say anything about what its subfields look like? Yes, we can
say something about one of its subfields. Let us prove this very startling and useful fact.
Theorem 6: Every field contains a subfield isomorphic to Q or to Z , for some prime number p.
p
Proof: Let F be a field. Define a function f : Z  F : f(n) = n.1 = 1 + 1 + .... + 1 (n times).
f is a ring homomorphism and Ker f = pZ, where p is the characteristic of F.
You know that char F = 0 or char F = p, a prime. So let us look at these two cases separately.

Case 1 (char F = 0): In this case f is one-one.  Z = f(Z). Thus, f(Z) is an integral domain contained
in the field F. Since F is a field, it will also contain the field of quotients of f(Z). This will be
isomorphic’ to the field of quotients of Z, i.e., Q. Thus, F has a subfield which is isomorphic to Q.

Case 2 (char F = p, for some prime p):
Since p is a prime number, Z/pZ is a field.
Also, by applying the Fundamental Theorem of Homomorphism to f, we get Z/pZ  f(Z). Thus,
f(Z) is isomorphic to Z and is contained in F. Hence, F has a subfield isomorphic to Z .
p
p
Let us Theorem 6 slightly. What it says is that:
Let F be a field.
(i) If char F = 0, then F has a subfield isomorphic to Q.

(ii) If char F = p, then P has a subfield isomorphic to Z.
Because of this property of Q arid Zp (where p is a prime number) we call these fields prime
fields.

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