Question

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.

Answer 1

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.