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Math Help - 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 n=p_1^{\alpha_1}\cdots p_m^{\alpha_m} be the prime factorization of n. Since n is not a square, there is some p_i with \alpha_i odd. So p_i divides n/p_i^{\alpha_i-1} but p_i^2 does not. So x^2-(n/p_i^{\alpha_i-1}) satisfies Eisenstein, which means it is irreducible over \mathbb{Q}. However, 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 p_i^{\alpha_i-1} is a square means \sqrt{n} is irrational.
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