Irreducible polynomial using Eisenstein Criterion? 10 points for informative answer!
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Irreducible polynomial using Eisenstein Criterion? 10 points for informative answer!

[From: ] [author: ] [Date: 11-05-22] [Hit: ]
.Please help me to understand and give a step by step answer.Thanks.-Part of the Eisenstein criterion says that coefficient of the highest degree term is NOT a multiple of the prime that youre considering. The criterion cant be applied be this way here because y+1 is both the prime youre considering andthe coefficient of the highest degree term (x^3).Try reversing the roles of the variables,......
Hello Experts,

I have the polynomial in Z[x,y]:

f(x,y) = x^3 *y + x^3 - x^2 * y - x^2 +x*y + x +y^2 + 2*y + 2.

I need to prove it's irreducible.
I want to use the Eisenstein Criterion where f(x,y) can be written like this:

x^3 * (y+1) - x^2 * (y+1) + x*(y+1) + 2(y+1) +y^2 in (Z[y])[x]

I know that y+1 is a prime in Z[y].

But using the criterion I am stuck....
Please help me to understand and give a step by step answer.
Thanks.

-
Part of the Eisenstein criterion says that coefficient of the highest degree term is NOT a multiple of the prime that you're considering. The criterion can't be applied be this way here because y+1 is both the prime you're considering and the coefficient of the highest degree term (x^3).

Try reversing the roles of the variables, where you consider this to be a polynomial in Z[x][y]. It has the form P(x)*y + Q(x), where P and Q are cubics in x. Some calculus will show you that each of these cubics has a non-integer root (and so is irreducible and so prime in Z[x]).
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