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Determine whether the polynomial in $Z[x]$ satisfies an Eisenstein criterion for irreducibility over $Q$. $x^ 2-12$ Which of the following is the right choice? Explain your answer.
(A) Yes, with $p=2$. (B) Yes, with $p=3$. (C) Yes, with $p=5$. (D) No.
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Eisenstein's criterion states that a polynomial $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+$ $a_1 x+a_0$ is irreducible over $Q$ if there exists a prime number $p$ such that:

1. $p$ divides all the coefficients $a_i$ for $0 \leq i \leq n-1$, but does not divide the leading coefficient $a_n$.

2. $p^2$ does not divide the constant term $a_0$.

In our case, the polynomial is $f(x)=x^2-12$.

Let's check each option:

(A) $p=2: 2$ divides the constant term -12 , but it does not divide the coefficient of $x^2$, which is 1 . Also, $2^2=4$ does not divide -12 . This case does not satisfy the first condition of Eisenstein's criterion.

(B) $p=3: 3$ divides the constant term -12 , but it does not divide the coefficient of $x^2$, which is 1 . Also, $3^2=9$ does not divide -12 . This case does not satisfy the first condition of Eisenstein's criterion.

(C) $p=5: 5$ does not divide the constant term -12 . This case does not satisfy the first condition of Eisenstein's criterion.

None of the given primes satisfy the conditions of Eisenstein's criterion for the polynomial $x^2-12$. Therefore, the correct choice is (D) No.
by Diamond (89,039 points)

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