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Apr 16, 2024
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MATH 6221 - Algebra I
G-sets. Orbits and stabilizers. Orbit decomposition formula. Permutation groups. Alternating groups. Simple groups. Composition series. Jordan-Holder Theorem. The Sylow Theorems. Fundamental theorem of abelian groups. Solvable and nilpotent groups. Rings of power series and Laurent series. Division rings. Prime and maximal ideals in a ring (not necessarily commutative). Nil radical. Rings of quotients of domains (not necessarily commutative). Artinian and Noetherian rings and modules. Hilbert Basis Theorem. Completely reducible modules. Semi-simple Artinian rings. Free, projective, and divisible modules. Tensor product of modules and algebras. Polynomial rings. Irreducible polynomials. Quotient rings. Eisenstein Criterion. Algebraic extension. Algebraically closed fields. Splitting fields. Normal and separable extensions. Finite fields. Fixed fields. Fundamental Theorem of Galois Theory. Solvability by radicals. Constructability by ruler and compass.
Requisites: MATH 5222
Credit Hours: 4
Repeat/Retake Information: May not be retaken.
Lecture/Lab Hours: 3.0 lecture
Grades: Eligible Grades: A-F,WP,WF,WN,FN,AU,I
Learning Outcomes:
- After completing both Algebra I and II students will acquire the background needed to attempt the Algebra comprehensive examinations.
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