Factorization (using eisenstein)

Hi,

Can someone please help me with the following question and explain to me what they are doing at each step:

In each case below, decide whether or not the given polynomial is irreducible over Q, justifying your answer in each case. If the polynomial is not irreducible , give its complete factorization into Q-irreducible factors.

i) (x^12) - 18(x^6) - 175

ii) (x^6) + (x^3) + 1

iii) (x^10) + 1


Thanks.

A polynomial is reducibble in Z[x] if and only if it is reducible in Q[x].

Quote:

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Originally Posted by jedoob

i) (x^12) - 18(x^6) - 175

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Begin this problem by writing,
y^2-18y-175=0 where y=x^6.
Find zeros and hence find the factors,
(y-25)(y+7)=0
Thus,
(x^3-25)(x^3+7)=0
Those two polyonomial are irreducible in Z[x].
Because,
x^3-25 has no zero (and has degree at most three).
x^3+7 has no zero (and has degree at most three).