Page 237 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Then n = deg f = deg g + deg h = m + r, and

a = b c + b c ... + b c  k = 0, 1 ..., n.
0
k
k
1
k-l
0
k
Now a = b c . We know that p | a . Thus, p | b c ,  p | b or p | c . Since p | a , p cannot divide
2
0
0 0
0
0
0
0
0 0
both b and c . Let us suppose that p | b and p k CJ
0
0
0
Now let us look at a,, = b, c,. Since p | a, we see that p | b and p | c . Thus, we see that for some
m
r
i, p | b . Let k be the least integer such that p | b . Note that 0 < k  m < n.
k
i
Therefore, p|a .
k
Since p|a and p|b , p | b , ..., p | b , we see that p(a – (b c + .... + b c ), i. e.,
0
1
k
0 k
k
k–1
k–1 1
p (b c . But p | b and p | c . So we reach a contradiction.
k 0
0
k
r Thus, f(x) must be irreducible in Z[x].
Let us illustrate the use of this criterion.
Example: Is 2x + 3x – 6x + 3x + 12 irreducible in Q[x]?
3
5
7
4
Solution: By looking at the coefficients we see that the prime number 3 satisfies the conditions
given in Eisenstein’s criterion. Therefore, the given polynomial is irreducible in Q[x].
Example: Let p be a prime number. Is Q[x]/<x – p > a field?
3
Solution: You know that for any field F, if f(x) is irreducible in F[x], then <f(x)> is a maximal
ideal of F[x].
Now, by Eisenstein’s criterion, x p is irreducible since p satisfies the conditions given in
3-
Theorem 3. Therefore, <x – p> is a maximal ideal of Q[x].
3
You also know that if R is a ring, and M is a maximal ideal of R, then R/M is a field.
Thus, Q[x] /<x – p> is a field.
3
Example: Let p be a prime number. Show that
f(x) = x + x + .... + x + 1 is irreducible in Z[x], f(x) is called the pth cyclotornic polynomial.
p-1
p-2
Solution: To start with, we would like you to note that f(x) = g(x) h(x) in Z[x] iff f(x + 1) = g(x + 1)
h(x + 1) in Z[x]. Thus, f(x) is irreducible in Z[x] iff f(x + l) is irreducible in Z[x].
p
x  1
Now, f(x) =
x 1

p
 f(x + 1) = x 1   1
x
1
= (x + C x + ... + C x + 1 – 1), (by the binomial theorem)
p
p
p-1
p
x 1 p-1
= x + px + C x + ... + C x + p.
p-1
p-2
p-3
p
p
p-2
2
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