Reducibility of a polynomial over all finite fields

[white karma]

Hi, I want to ask if there's other solution for (b). I used the legendre symbols to prove at least one of 2,7,14 is a quadratic residue, and the rest is trivial

Thank you so much if you could provide other ideas or related problems.

[opal marsh]

Do you need help ?

If so this isn’t the right channel

But to answer your question yes you can solve b differently

Actually you can use (c) to solve (b)

because P being irreducible mod (p) implies that Gal(K/Q) contains a 4-cycle (if you see it as a subgroup of S_4) notably it contains an element of order 4

but you can see that this isn’t possible

Up to you to see why