Let be a monic polynomial and let be prime.
- Show that if is irreducible in , then is irreducible in .
- Show via a counterexample that the converse to part (a) is false.
- Use this to decide whether the following are irreducible over : , .
Suppose is irreducible mod . If is a factorization of over , then by Gauss’ lemma it is reducible over . (That is, we may assume that . Then factors in . Now without loss of generality, is a unit. That is, has degree 1. If the degree of is greater than 1, then the leading term of is divisible by . In particular, divides the leading term of , a contradiction since is monic. So has degree 1, and hence is a unit in . Thus is irreducible.
Note that is irreducible over by Eisenstein’s criterion. However, mod 2 this polynomial is congruent to , which is clearly reducible. So the converse does not hold in general.
We claim that is irreducible mod 3. To see this, note that mod 3. By Fermat’s little theorem, mod 3 for all , so that under the evaluation homomorphism we have mod 3 for all . In particular has no roots mod 3, hence no linear factors. Any factorization of must include a linear factor; thus is irreducible mod 3. Hence is irreducible in .
We claim that is irreducible mod 2. Note that has no linear factors, since (as we can easily see) has no roots in . Thus if were to be reducible, it would factor as a product of two irreducible quadratics. Note that there are precisely four quadratic polynomials mod 2: , , , and . We claim that the last of these is irreducible. Indeed if is reducible mod 2, then it must have a linear factor and hence a root. However, we see this is not the case. Thus is the only irreducible quadratic polynomial in , and certainly mod 2. So is irreducible mod 2, and thus must be irreducible over .