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Catalog Description:
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Groups, rings, fields, homomorphisms, and quotient structures. Prerequisite: MATH
261. (Offered fall semester.)
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Required Course Materials:
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John B. Fraleigh, A First Course in Abstract Algebra,
7th edition, Addison Wesley, Boston,
New York, 2002.
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Course Coordinator:
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Angela C. Hare, Ph.D., Associate Professor of Mathematics
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Course Audience:
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Juniors and seniors majoring in Mathematics.
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Course Objectives:
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- To gain a good working knowledge of the standard basic terminology,
definitions and techniques of modern algebra.
- To have an in depth encounter with axiomatic mathematics with attention to
concise definitions and properly constructed proofs.
- To develop skill in concise, clear and logical presentation of problem solutions
and proofs both orally and in writing.
- To experience the interplay between attempts to find a proof and attempts to
find a counterexample which eventually culminates in determination of truth or
falsehood of a conjecture.
- To carry out some of the significant computational exercises of this branch of
mathematics, for example construction of the multiplication table for a finite
field.
- To gain familiarity with the use of the
program Exploring Small Groups.
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Prerequisites:
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- Abstract vector spaces.
- Function concept including real-valued function, linear transformations of
vector spaces, determinant as a mapping, etc.
- Polynomials and rational root test.
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Topics:
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- Groups: definition, subgroups, order, permutation groups, Lagrange's
theorem, homomorphisms, normal subgroups, quotient (factor) groups, direct
product of groups.
- Rings: definition, subrings, integral domains, ideals and factor rings, direst
products, homomorphisms, quotient rings, rings of polynomials,
factorization.
- Fields: definition, extension fields, finite fields.
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Resources:
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- Classrooms with PC access to network.
- Computer labs (Frey 145, 166) for student use of Exploring Small Groups
software.
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Pedagogy:
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This course focuses on the deductive method and mathematical proof. The proof writing exercises are a distinguishing feature.
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