Prove that two given polynomials are irreducible over the Gaussian rationals

Show that p(x) = x^3-2 and q(x) = x^3-3 are irreducible over F = \mathbb{Q}(i).


Recall that the Gaussian integers \mathbb{Z}[i] are a Unique Factorization Domain, and that \mathbb{Z}[i] \subseteq \mathbb{Q}(i). We claim that \mathbb{Q}(i) is the field of fractions of \mathbb{Z}[i]. Indeed, if F is a field containing \mathbb{Z}[i], then F also contains any Gaussian rational since we can write \frac{a}{b} + \frac{c}{d}i = \frac{1}{bd}(ad + bci) with a,b,c,d \in \mathbb{Z}. By the universal property of fields of fractions, \mathbb{Q}(i) is the field of fractions of \mathbb{Z}[i].

Recall that 1+i and 3 are irreducible in \mathbb{Z}[i]. So Eisenstein’s criterion applies, showing that p and q are irreducible over \mathbb{Q}(i).

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Comments

  • cryan  On April 28, 2012 at 9:21 pm

    Should be q(x) = x^3 – 3.

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