Given n ∈N and p prime number and consider the polynomial f (x) = xn (xn-2)+1-p...

Given n ∈N and p prime number and consider the polynomial f (x) = xⁿ(xⁿ-2)+1-p

1)Prove that f (x) is irreducible in Q [x]

2) If n = 1 and p = 3, find Q [x] / f (x))

3) Show that indeed Q [x] / (f (x)) is a field in the previous paragraph

PLEASE answer all subsections

Expert Solution

First to prove the irreducibility of f, we will use the property of a prime number, and further , we wil prove field axiom existence of inverse.

And hence we can easily conclude that

Q[x] / (f(x)) does not have any zero divisor elements.

Thus , this is a field.

Colby Messinger answered 1 year ago

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Question

Given n ∈N and p prime number and consider the polynomial f (x) = xn (xn-2)+1-p...

Solutions

Expert Solution

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