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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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Solution Summary

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

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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 ...

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