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MATHMAA

# Eisenstein_Criterion

Definition of Eisenstein Criterion :-

Suppose F(x) = $$a_nx^n$$ + $$a_{n-1}$$ + -------------- + $$a_1x$$ + $$a_0$$  is a polynomial with integer coefficients .

Let 'p' be a prime number such that

1.    p divides each $$a_i$$ for $$i \neq 0$$

2.   p does not divide $$a_n$$, and

3.   $$p^2$$ does not divide $$a_0$$

Then F(x) is irreducible over the rational number. It will also be irreducible over integers,  unless all its coefficients have a nontrivial factor in common.

2) Suppose that P is a prime number prove that \(