Unknown

3.1. Irreducible Polynomials 83

11. Let p(t) be a manic polynomial over Z. Show that, if r is a rational
    root of p(t), then r must be an integer.


12. The Eisenstein Cri2erion. Given a polynomial over Z, it is usually
    difficult to decide immediately whether it is irreducible. Neverthe-
    less, it is possible to determine conditions which will decide the issue
    in a large number of cases and which will permit the easy construc-
    tion of irreducible polynomials. One important result is the following
    Eisendein Criterion:

Suppose that the polynomial h(t) = c,t” +c,-It”-’ +. + .+

clt + co has integer coefficients ci and that there is a prime

integer p for which

(a) p is not a divisor of the leading coefficient c,;

(b) p is a divisor of every other coefficient CO, cl,... ,cn-1;

(c) p2 does not divide the constant coefficient CO.

Then the polynomial q(t) is irreducible over Z.

To get a handle on the proof of this, consider the special case that

h(t) is a cubic satisfying the conditions. Suppose that q(t) is reducible.

Then it can be written in the form

cst3 + czt2 + clt + co = (bit + bo)(azt2 + alt + a,~).

Write out the ci in terms of the ai and bi, and argue that p must divide

exactly one of a0 and bo, say ao. Deduce that p must accordingly

divide al and ~2, yielding a contradiction. Now give a proof of the

criterion for polynomials of arbitrary degree.

13. Use the Eisenstein Criterion with a suitable prime to show that 2t4 +
    21t3 - 6t2 + 9t - 3 is irreducible over Z.


14. Find a linear polynomial over Z which does not satisfy the Eisenstein
    Criterion.


15. Show that the polynomial t2 + t + 1 does not satisfy the Eisenstein
    Criterion for any prime, yet is irreducible over Z.


16. Let h(t) = cZm+ltzm+’ + cg,t”” + ... + clt + CO be a polynomial of
    odd degree 2m + 1 2 3 over Z. Suppose that, for some prime p,

(a) P$cz~+~;

(b) plci (m+lIi-Q2m);

Cc> P2 I cj (0^5 j^5 ml;

(4 p34’co.