Page 241 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Similarly, each of b , b , ...., b has p choices. And, corresponding to each of these choices we get
n
2
3
a distinct element of F. Thus, the number of elements in F is p × p × ... × p (n times) = p .
n
The utility of this result is something similar to that of Lagrange’s theorem. Using this result we
know that, for instance, no field of order 26 exists. But does a field of order 25 exist? Does
Theorem 7 answer this question? It only says that a field of order 25 can exist. But it does not say
that it does exist. The following exciting result, the proof of which is beyond the scope of this
course, gives us the required answer. This result was obtained by the American mathematician
E.H. Moore in 1893.

Theorem 8: For any prime number p and n  N, there exists a field with pn elements. Moreover,
any two finite fields having the same number of elements are isomorphic.

Self Assessment

1. If f(x)  Z(x) is irreducible over Q[x]. Then it is .............. in Q[x].
(a) reducible (b) irreducible

(c) direct (d) finite
2. A non-empty subsets of a field F is called a .............. of F it is a field with respect to the
operation on F.

(a) subfield (b) field domain
(c) range field (d) extension
3. Every field contains a subfield is o morphic to Q or to Z for r some .............. P.
p
(a) prime (b) finite
(c) infinite (d) external
4. Let F be a finite having of elements and characteristics P, then q = .............., some positive
integer n.
(a) p -1 (b) p n
(c) xp n (d) p.x p
5. For any prime number P and n  N, then exists a field with Pn elements. Move over, and
two .............. fields having the same number of elements are isomorphic.
(a) infinite (b) finite
(c) direct (d) extension

24.3 Summary

Gauss lemma, i.e., the product of primitive polynomials is primitive.

For any n  N, we can obtain an irreducible polynomial over Q of degree n.

Definitions and examples of subfields and field extensions.

Different ways of obtaining field extensions of a field F from F[x].


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