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Thread: Show a polynomial satisfies Eisenstein's criterion

  1. #1
    Junior Member
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    Show a polynomial satisfies Eisenstein's criterion

    Show that if the positive integer n is not a square, then for some p and some integer k, the polynomial (x^2 - [n/p^(2k)]) satisfies Eisenstein's criterion. Conclude that
    n^(1/2) is not a rational number.

    I know that Eisenstein's criterion states:
    Let f(x) = anx^n + ... + a1x + a0 have integer coeffiecients. If there exists a prime p such that an-1 an-2 ≡ ... ≡ a1 ≡ a0 mod p, but an is not ≡ to 0 mod p and a0 is not ≡ to 0 mod p^2. Then f is irreducible over Q.

    Any help would be greatly appreciated.
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  2. #2
    Senior Member
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    Let $\displaystyle n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$ be the prime factorization of $\displaystyle n$. Since $\displaystyle n$ is not a square, there is some $\displaystyle p_i$ with $\displaystyle \alpha_i$ odd. So $\displaystyle p_i$ divides $\displaystyle n/p_i^{\alpha_i-1}$ but $\displaystyle p_i^2$ does not. So $\displaystyle x^2-(n/p_i^{\alpha_i-1})$ satisfies Eisenstein, which means it is irreducible over $\displaystyle \mathbb{Q}$. However, $\displaystyle x^2-(n/p_i^{\alpha_i-1})=(x-\sqrt{n/p_i^{\alpha_i-1}})(x+\sqrt{n/p_i^{\alpha_i-1}})$, which since $\displaystyle p_i^{\alpha_i-1}$ is a square means $\displaystyle \sqrt{n}$ is irrational.
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