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Department of Mathematics
MATH338 - Algebra IIIB
Unit Syllabus
Galois Theory:
- Polynomials, test for irreducibility,
numbers of real zeros of rational polynomials using exact methods.
- Ruler and compass constructions.
- Number fields, constructible number
fields, field extensions, minimum polynomials.
- The solution of the quadratic and the
cubic from an advanced standpoint.
- Radical extensions, field automorphisms,
Galois groups, extending and restricting automorphisms.
- Solubility of polynomials by radicals.
- Insolubility of the quintic.
Ring Theory:
- Elementary properties of rings, ring
homomorphisms, localization of integral domains, unique factorization.
- Modules, direct sums and products, semisimple
modules, chain conditions, modules with finite length, tensor
products, modules over principal ideal domains.
- Structure of non-commutative rings,
prime and primitive ideals, the Jacobson radical, semisimple
Artinian rings.
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