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Problems in Galois Theory Involving a Cubic Polynomial

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Let f be a cubic monic polynomial and char K not 3.
a. Show how to make a change of variables x' = x - lamda in f (x) = 0 to reduce to a monic equation where the coefficient of x^2 is zero
b. Suppose K = R. Let D be the discriminant of f. Prove that f has one real root if D < 0 and three real roots if D > or equal to 0.
c. Let f be irreducible, char K not 2, 3. Prove that the Galois group of f is cyclic of order 3 if D is a square in K, and is the symmetric group S_3 otherwise.

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** Please see the attachment for the complete solution response **

We let be a monic cubic polynomial over a field K with (please see the attached file).
(a) We wish to make a change of variables such that has no quadratic term. We have (please see the attached file)

whence the ...

Solution Summary

In this solution, we solve three problems in Galois theory involving a cubic polynomial.