# 2013-23 Polynomials with rational zeros

Find all polynomials $$P(x) = a_n x^n + \cdots + a_1 x + a_0$$ satisfying (i) $$a_n \neq 0$$, (ii) $$(a_0, a_1, \cdots, a_n)$$ is a permutation of $$(0, 1, \cdots, n)$$, and (iii) all zeros of $$P(x)$$ are rational.

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Find all field automorphisms of the field of real numbers $$\mathbb{R}$$. (A field automorphism of a field $$F$$ is a bijective map $$\sigma : F \to F$$ that preserves all of $$F$$’s algebraic properties.)