Page 242 - DMTH403_ABSTRACT_ALGEBRA

Unit 24: Irreducibility and Field Extensions

Eisenstein’s irreducibility criterion for polynomials over Z and Q. This states that if Notes

f(x) = a + a, x + . . . + a x  Z[x] and there is a prime p  Z such that
n
0 n
p | a  i = 0 , 1 . ..., n – 1.
 i
p | a, and

p | a ,
 0
then f(x) is irreducible over Z’ (and hence over Q).
Every field contains a subfield isomorphic to a prime field.

The prime fields are Q or Zp, for some prime p.
The number of elements in a finite field F is pn’, where char F = p and dim z F = n.
 p
Given a prime number p’and n  N, there exists a field containing p elements. Any two
n

finite fields with the same number of elements are isomorphic.
If F is a finite field with pn elements, then x p n  x is a product of pn linear polynomials

over F.

24.4 Keywords

Eisenstein’s Criterion: Let f(x) = a + a x + ... + a,,x  Z[x]. Suppose that for some prime number
n
l
0
p; (i) P | a , (ii) p | a , p | a ,...p|a n–1
1
0
n
Subfield of F: A non-empty subset S of a field F is called a subfield of F if it is a field with respect
to the operations on F. If S$F, then S is galled a proper subfield of F.
24.5 Review Questions
1. What are the contents of the following polynomials over Z?

(a) 1 + x + x + x + x 4 (b) 7x – 7
4
2
3
(c) 5(2x – 1) (x + 2)
2
2. Prove that any polynomial f(x)  Z[x] can be written as dg(x), where d is the content of f(x)
and g(x) is a primitive polynomial.
3. For any n  N and prime number p, show that x – p is irreducible over Q[x]. Note that this
n
shows us that we can obtain irreducible polynomials of any degree over Q[x].
4. If a + a x + ... + a, x  Z[x] is irreducible in Q[x], can you always find a prime p that satisfied
n
0
1
the conditions (i), (ii) and (iii) of Theorem 3?
5. Which of the following elements of Z[x] are irreducible over Q?
(a) x – 12 (b) 8x + 6x – 9x + 24
2
2
3
(c) 5x + 1
6. Let p be a prime: integer. Let a be a non-zero non-unit square-free integer, i.e., b |a for
2
any b  Z. Show that Z[x]/<x + a> is an integral domain.
p
7. Show that x  a Z [x] is not irreducible for any a Z.
p

p

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