# Polynomial Question: Galois group is S7

Find a polynomial of degree 7 over Q whose Galois group is S7.

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Find a polynomial of degree 7 over Q whose Galois group is S7

Solution: Let (please see the attached file) be an irreducible polynomial of degree 7 over (please see the attached file) and let (please see the attached file) be the splitting field of (please see the attached file). Then the Galois group of (please see the attached file) is the group (please see the attached file) such that (please see the attached file). Then if (please see the attached file) is a root of (please see the attached file) then (please see the attached file). So 7 divides the order of the Galois group (please see the attached file) of (please see the attached file). ...

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This solution explains how to find a polynomial for a given problem.