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Let α = 4√ 3 (∈ R), and consider the homomorphism ψα : Q[x] → R...

Let α = 4√ 3 (∈ R), and
consider the homomorphism
ψα : Q[x] → R
f(x) → f(α).

(a) Prove that irr(α, Q) = x^4 −3

(b) Prove that Ker(ψα) = <x^4 −3>

(c) By applying the Fundamental Homomorphism Theorem to ψα, prove that
L ={a0+a1α+a2α2+a3α3 | a0, a1, a2, a3 ∈ Q }is the smallest subfield of R containing α.
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