Galois Theory / Solvability by Radicals, Eisenstein's Criteria and Irreducibility

Related BrainMass Solutions

• Irreducible Polynomials

  We can use the Eisenstein's Irreducibility Criterion. Let p=2, then p is a prime. We check all the coefficients of f(x) and find the following facts.

• Eisenstein's Criterion

  Use the fact that to show that , and so find coefficients such that . Hence show that is irreducible over by using Eisenstein's criterion. Please see the attached file. This solution provides a proof of irreducibility.

• Ireducibility

  This is a proof regarding irreducibility and a polynomial ring.

• Polynomial Question: Galois group is S7

  So (please see the attached file) is irreducible over (please see the attached file) by the Eisenstein Irreducibility Criterion. We can verify graphically that this polynomial has exactly five real roots.

• Fundamental Theorem of Galois Theory

  334133 Fundamental Theorem of Galois Theory Suppose K is a Galois extension of F of degree p^n for some prime p and some n >= 1. Show that there are Galois extensions of F contained in K of degree p and p^(n-1).

• Problems in Galois Theory Involving a Cubic Polynomial

  450067 Problems in Galois Theory Involving a Cubic PolynomialK Let f be a cubic monic polynomial and char K not 3. a.

• Irreducible Polynomials

  This polynomial is irreducible over Q ( a , b ) , hence ( by Hilbert's irreducibility theorem ) is irreducible over Q for infinitely many specializations of a and b into Q .

• Listing Subgroups and Subfields

  181582 Listing Subgroups and Subfields Please list the subgroups and subfields detailed in the attachment and then use the Fundamental Theorem of Galois Theory to ensure you've got them all. Please see the attachment.

• Problems in Galois Theory

  To show that this is all of them, note that by the fundamental theorem of Galois theory, the set of subfields of K is isomorphic to the set of subgroups of G(Gal(K/Q)).