Question

Prove that f(t)=t^7+10t^2−5 is irreducible over Q

Answer 1

\[\Large t^7+10t^2−5 = 0\]Not quite sure how to prove this has no rational solutions.

Answer 2

Oh why not just do the rational zeroes test? http://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml

There's only two (well four with plus/minus). Check all possible factors p/q:
\[\Large \pm 1, \pm \frac{ 1 }{ 5 }\]

If there are no rational zeroes, then the polynomial is irreducible over the rational numbers.

Answer 3

use the rational roots theorem