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**page929** Use Eisenstein's criterion to show taht each of these polynomials is irreducible over the field of rational numbers. (You may need to make a substitution).

a) x^4 - 12x^2 + 18x - 24

b) 4x^3 - 15x^2 + 60x + 180

c) 2x^10 - 25x^3 + 10x^2 - 30

d) x^2 + 2x - 5 (substitute x-1 or x+1)

Here is what I have so far:

a) p = 2 or 3

when p = 2, then -24 ≡ 0 mod 2^2

and when p = 3, then -24 ≡ 3 mod 3^2

so, p = 3

also, 1 ≡ 0 mod 3 is not true

b) p = 3 or 5

when p = 3, then 180 ≡ 0 mod 3^2

and when p = 5, then 180 ≡ 5 mod 5^2

so, p = 5

also, 4 ≡ 0 mod 5 is not true

c) p = 5

so, -30 ≡ 6 mod 5^2

also, 2 ≡ 0 mod 5 is not true

d) I am not sure

Does what I have done look correct? Any help for part d would be greatly appreciated.