What makes a polynomial irreducible?
I'm not seeing anyone touch an the real (pun intended) answer.
Polynomials are never irreducible. Over the complex numbers.
They are only irreducible over the reals (or rationals, integers etc.).
A polynomial that is irreducible over the reals is one that has strictly complex roots. (note this will be an even degree polynomial. why? )
what are the roots of $x^4 + 64$ ?
w/o going into detail these are $x=\pm 2 \pm 2 i$
i.e. all the roots are complex
But $x^4 + 64$ factors into $(x^2 - 4x + 8)(x^2 + 4x + 8).$ That means it is
reducible over the reals. In particular, it is reducible over the integers.
** From above in the lower quote box. This polynomial has strictly complex
roots, but it is not irreducible.