Doubt in'Finding irreducible polynomial'

Question

This is from Artin Algebra Sec 15.4:

I can't understand how did we conclude that $x^4-10x^2+1$ is irredducible over $\Bbb Q$ from the discussion of para above example 15.4.4.

Answer 1

Well, $\gamma$ is a zero of $x^4-10x^2+1$ over the rationals and this is the lowest degree polynomial with this property. So it must be irreducible.