Math 4030, Foundations of Algebra
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Prerequisites: Math 2220 and one of Math 2210 or 1260 or 1280. Some knowledge of Linear Algebra would be
helpful.
Textbook: Lecture notes written for this course by Aaron Bertram. They are available at:
http://www.math.utah.edu/∼bertram/courses/4030/
Course Description: We will study some of the basic algebraic structures that arise in mathematics, going
from concrete examples to more general and abstract concepts. First we will study various concepts of numbers:
natural numbers and the principle of induction, then integers and rational numbers, then real and complex
numbers.
The main part of the course will be on polynomials and their roots. This will bring us to algebraic numbers,
and to the concepts of rings and fields. We will see finite fields and how they help us understand polynomials.
The culmination of the course will be the application of these concepts to the proof of the impossibility of certain
ruler and compass constructions, for example, the impossibility of trisecting certain angles by ruler and compass.
Course Objectives: At the end of this course, the students will:
1. Know definitions and examples of rings (integers) and fields (rational and real numbers).
2. Demonstrate understanding of field axioms.
3. Understand rational numbers as equivalence classes of integer fractions (ordered pairs of integers with
appropriate constraints), define the operation of addition and multiplication and prove that so defined the
set of rational numbers with the given operations forms a field.
4. Define real numbers as limits of sequences of real numbers, define operations on real numbers using geometric notions and be able to prove that a set of real numbers with so defined operations forms a field.
5. Students will be able to convert between repeating decimals and fractional representations of rational
numbers. They will be able to explain which rational numbers correspond to terminating decimals.
6. Use the isomorphism between C and R2 and polar coordinates to offer geometric interpretation for multiplication of complex numbers.
7. Find n-th roots of any complex number.
8. Know how to extend the ideas from rational and real numbers to the field of polynomials and rational
functions.
9. Be able to articulate how the familiar notion of degree of a polynomial and Euclidean algorithm generalize
to a degree function and Euclidean domains.
10. Prove and use the rational root test.
11. Implement some tests to determine whether a polynomial is prime in Q[x].
12. Understand the constructible number theorem and know how to construct square roots of constructible
numbers.
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Evaluation Methods and Criteria:
• Homework and Quizzes: Every week I will assign homework problems. These are to be worked on
schedule, but not to be handed in. Every Friday, beginning August 31, I will give a 20 minute quiz on
the homework on material discussed by the previous Monday. Please make sure that you can do all the
homework problems, and get all the help you need. You are encouraged to ask questions in class, or during
office hours. Since I will drop the lowest three quiz scores, there will be no make-ups for quizzes.
• Midterms: Two midterm exams on September 28 and November 9.
• Final: Comprehensive final exam on Wednesday December 12, 10:30–12:30.
• Grading:
Quizzes, drop lowest 3:
Two Midterm Exams:
Final exam:
30 %
40 %
30 %
Important dates: Last day to add classes Sunday August 26. Last day to drop (delete) classes: Wednesday,
August 29. Last day to withdraw from classes: Friday, October 19.
ADA: The University of Utah seeks to provide equal access to its programs, services and activities for people
with disabilities. If you will need accommodations in the class, reasonable prior notice needs to be given to the
Center for Disability Services, 162 Olpin Union Building, 581-5020 (V/TDD). CDS will work with you and the
instructor to make arrangements for accommodations. All information in this course can be made available in
alternative format with prior notification to the Center for Disability Services.
Schedule of Topics and Due Dates:
This list of topics closely follows the course notes. Each topic will take between one and two weeks. Some topics
in the first chapter overlap with Math 2200 and will be covered more quickly. The emphasis would be in the
later chapters. If time permits, we will study the last few topics in more detail. In that case additional notes
will be provided.
1. Numbers.
(a) Induction and the natural numbers.
(b) Additive inverses and the ring of integers.
(c) Equivalence classes and the field of rational numbers.
(d) Limits and the real numbers.
(e) The complex numbers and polar coordinates.
2. Polynomials
(a) The arithmetic of polynomials.
(b) Euclidean domains and fundamental theorem of arithmetic.
(c) Roots of polynomials and algebraic numbers.
(d) Clock arithmetic and finite fields.
3. Matrices.
(a) Basic linear algebra, characteristic polynomial of a matrix.
(b) Constructible numbers.
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