Midterm Conceptual Review
Math 558: Introductory Modern Algebra, Fall 2015
As always, feel free to come to office hours with questions!
Chapter 1.
(a) Concepts: equivalence relation, equivalence class
(b) Goals: identify whether a binary relation is an equivalence relation, determine the equivalence class of an element in a set with an equivalence relation
(c) Homework problems: 1 - 3
Chapter 2. (2A - 2D)
(a) Concepts: Principle of Mathematical Induction, Principle of Complete Mathematical
Induction, Principle of Infinite Descent, Well-ordering Principle, binomial coefficient, Binomial Theorem
(b) Goals: Decide which principle above to apply (if applicable), and use it to prove a statement, use induction to prove statements about binomial coefficients
(c) Homework problems: 1, 8, 13, 20, 23, 27, 30-33, 37
(d) Additional practice problems: 2, 22, 26, 35
Chapter 3. (3A - 3E)
(a) Concepts: Division Algorithm, divisor/divides, greatest common divisor, Euclidean Algorithm, Bézout’s theorem, linear Diophantine equation
(b) Goals: Implement the Division and Euclidean Algorithms, use the Euclidean Algorithm to
find the appropriate values in Bézout’s theorem, decide whether a linear Diophantine equation has a solution (and if it does, find all solutions), show statements on divisors/integers
using the concepts above
(c) Homework problems: 3-5, 23, 25, 29, 33, 39, 40, 60, 62, 65
(d) Additional practice problems: 26, 28, 30, 34, 36, 37, 46, 61
Chapter 4. (4A - 4C)
(a) Concepts: Fundamental Theorem of Arithmetic (FTA), prime, least common multiple
(b) Goals: Prove statements about integers using the FTA (including that certain numbers
are not rational)
(c) Homework problems: 2, 3, 5, 10, 13, 21, 23, 32, 37
(d) Additional practice problems: 1, 4, 11, 18, 27, 38
Chapter 5. (5A - 5B, 5F)
(a) Concepts: congruence modulo m, least non-negative residue, linear equations modulo m
(b) Goals: Translate back and forth between congruences and integer equations using the
definition of congruence modulo m, show properties of congruence modulo m, solve linear
equations modulo an integer m
Math 558: Introductory Modern Algebra, Fall 2015, Fall 2015
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(c) Homework problems: 4-6, 11-15, 19, 22, 44, 48-50
(d) Additional practice problems: 7, 16, 20, 45, 47
Chapter 6. (6A - 6D)
(a) Concepts: congruence classes, Z/mZ, arithmetic modulo m, complete sets of representative, primitive root, Primitive Root Theorem
(b) Goals: perform arithmetic and compute inverses in Z/mZ, find primitive roots, solve
equations in Z/mZ
(c) Homework problems: 6, 8-10, 18, 22, 34
(d) Additional practice problems: 17, 23, 33
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