Math 414 syllabus

Algebra II (W)

Course Description: A continuation of MA 413 focusing on rings and fields. Topics include rings, ideals, integral domains, fields and extension fields. Geometric constructions and Galois theory are introduced.
Prerequisites: MA 413 Minimum Grade of C and (EH 102 Minimum Grade of C or EH 105 Minimum Grade of C).

Suggested Text: A Book of Abstract Algebra (Second Edition) by Charles C. Pinter, Dover Publications, Inc.
Coverage: Selection of topics from chapters 24 – 27 and chapters 29 – 33.

Learning outcomes: Upon the successful completion of the course a student will:
write short proofs (direct, by contradiction, and using the contrapositive),
disprove algebraic statements by finding a counterexample,\
state, justify, and apply basic properties of rings and fields,
verify that a given subset of a field is a subfield,
state and apply the division algorithm for polynomials,
compute in the fields of rational functions,
compute with polynomials over finite fields,
find roots of polynomials over finite fields,
decide whether a polynomial is irreducible over a given field,
state and prove the Unique Factorization Theorem,
state, prove, and apply Eisenstein’s Irreducibility Criterion,
decide whether elements in a field extension are algebraic or transcendental,
compute the minimal polynomial of an algebraic element over a given field,
compute the degree of a field extension,
prove that the set of constructible numbers is a field,
construct points in the plane by ruler and compass,
prove that doubling the cube, trisecting an arbitrary angle, and squaring the circle are impossible with ruler and compass,
determine the Galois group of a given polynomial,
state and prove the Fundamental Theorem of Galois Theory,
prove the existence of polynomials of fifth degree that are not solvable by radicals.