Using Eisenstein to prove irreducibility in Q

[stgermaine]

Homework Statement

Use Eisenstein's criterion to show that 2*x^4 - 8x^2 + 3 is irreducible in Q[x]

Homework Equations

Eisenstein's criterion states that a polynomial is irreducible in Q[x] if the following three conditions are met for a prime p.
(i) p divides all coefficients except a_n and a_0.
(ii) p does not divide a_n
(iii) p^2 does not divide a_0

The Attempt at a Solution

The only prime that divides all coefficients except a_n and a_0 is 2. However, 2 does divide a_n, but its square does not divide a_0.

 

[lippyka]

Thus, 2 meets the conditions of Eisenstein's criterion and so 2*x^4 - 8x^2 + 3 is irreducible in Q[x].