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Galois Theory - pages.uoregon.edu

Galois TheoryRichard KochDecember 19, 2017 Contents1 The Extension Problem; Simple Groups .. An Isomorphism Lemma .. Jordan Holder .. The Symmetric and Alternating Groups ..82 The Quadratic, Cubic, and Quartic The Quadratic Formula .. The Cubic Formula .. The Quartic Formula ..153 Field Extensions and Root Motivation for Field Theory .. Fields .. An Important Example .. Extension Fields .. Algebraic Extensions; Root Fields .. Irreducible Polynomials overQ.. The Degree of a Field Extension .. Existence of Root Fields .. Isomorphism and Uniqueness .. Putting It All Together ..294 Splitting FactoringP.. The Splitting Field.

this quotient information which is important in Galois theory. In the previous section, we listed the three groups of order four obtained by extending Z 4 by Z 2. Notice that the simple quotients of all three groups are Z 2;Z 2;Z 2. So in this case, extension information is de nitely thrown away.

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