618-321 Algebra

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Objectives:

On completion of this subject, students should:

Comprehend:

• the concepts of unique factorisation domains;

• fields of fractions; modules;

• algorithmic nature of the structure theorem for modules over Principal Ideal Domains when specialised to Euclidean Domains;

• Galois correspondence; unsolvability in general of equations by radicals.

Have developed:

• the ability to find the structure of finitely generated abelian groups from their presentations;

• the ability to test polynomials of low degree for irreducibility; an understanding of the impossibility of trisecting an angle by ruler and compass;

• the ability to calculate Galois groups of equations in special cases.

Appreciate:

• the structure of special rings like Principal Ideal Rings;

• the possibility of relating problems in different areas by correspondences like Galois correspondence;

• that certain problems are not solvable and that it is possible to prove that they are not solvable in some interesting cases.

Content:

Modules over principal ideal domains: review of basic ring theory; ideals, quotients, the homomorphism theorems, prime and maximal ideals; integral domains and the field of quotients; Euclidean domains and principal ideal domains; definition and examples of modules; submodules, homomorphisms of modules, quotient modules; free modules and bases; structure of a finitely generated module over a principal ideal domain; applications to abelian groups. Field Theory: field extensions and their construction; the degree of a field extension; ruler and compass constructions; splitting fields; the Galois group of a field extension; the fundamental theorem of Galois theory.

Assessment: