Download Final Exam conceptual review

Final Exam conceptual review
Math 558: Introductory Modern Algebra, Fall 2016
• The final exam is cumulative, although there will be a bit more focus on material
after the midterm, as you have not be tested on it thus far.
• We have covered sections 1, 2A - 2D, 3A - 3E, 4A - 4C, 5A - 5B, 5F, 6A - 6F, 7A, 7C 7D, 9A - 9B, 10A, 12A, 12D, 13A - 13B, 14A - 14C, 15D, 15F, 17A - 17B, and 23B.
• The conceptual review sheets for Midterms 1 and 2 is also posted on the course
webpage.
• Make sure to study the problems assigned after the midterm, including the final homework problems (whose solutions are posted).
• You are responsible for knowing the precise statement of every important theorem
and definition that appears in the Daily Update.
• Like the midterm, the exam will be composed of a variety of both proof-based and
computational problems.
• After the traditional conceptual review below, we give some practice problems (some
challenging!). These do not cover a comprehensive list of topics! Make sure to
do all homework problems as well.
As always, feel free to come to office hours with questions!
Chapter 12. (12A, 12D)
(a) Concepts: Chinese Remainder Theorem (CRT),
(b) Goals: Decide whether a system of congruence equations can be solved, with justification,
and solve the system if possible using the CRT; state the CRT in terms of isomorphisms;
show that the homomorphism ψ in the CRT is, or is not, an isomorphism; use the CRT
in terms of isomorphism to solve a system of congruence equations; determine whether
a function between rings is a homomorphism/isomorphism; use the definition of a homomorphism
(c) Homework problems: #1 - 3, 6, 11, 12, 47, 48, 49, 50; Daily Update #5 - 9
(d) Additional practice problems: #7, 10, 11
Chapter 13. (13A - 13B)
(a) Concepts: Polynomial with coefficients in a commutative ring, the set R[x] of all polynomials with coefficients in a commutative ring R, the degree of a polynomial
(b) Goals: Do arithmetic with polynomials, apply the proposition on the degree of the product
of polynomials, find the units and the zero divisors of a polynomial ring
(c) Homework problems: #1, 2, 3, 4, 5
Math 558: Introductory Modern Algebra, Fall 2015, Fall 2015
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(d) Additional practice problems: #6(i, iii)
Chapter 14. (14A - 14C)
(a) Concepts: Division Algorithm for polynomials, the Fundamental Theorem of Arithmetic
(unique factorization) for polynomials, associate polynomials, the Root Theorem, the Remainder Theorem, D’Alembert’s Theorem, (a/the) greatest common divisor of two polynomials, the Euclidean Algorithm for polynomials, irreducible and reducible polynomials
(b) Goals: Apply the Division Algorithm for polynomials, apply the Remainder Theorem,
apply the Root Theorem, apply the Euclidean Algorithm for polynomials to find a/the
greatest common divisor of two polynomials, find polynomials satisfying Bézout’s Theorem
for polynomials using the back-substitution method, show that two polynomials are relatively prime, apply the methods thus far to decide whether a polynomial is irreducible or
reducible (including the Root Theorem), factor a polynomial into a product of irreducible
polynomials, find the greatest common divisor of two polynomials by factoring each into
a product of irreducible polynomials and identifying all associate pairs
(c) Homework problems: #4, 7, 8, 10, 14, 15, 22, 25, 26, 28, 37, 42, 43, 46; Daily Update
#10
(d) Additional practice problems: #1, 6, 16, 19, 21, 27, 30, 39, 41, 44, 47
Chapter 15. (15D, 15F)
(a) Concepts: The Fundamental Theorem of Algebra
(b) Goals: Check whether a given complex number is a root of a polynomial in C[x], factor
polynomials in R[x] into irreducible polynomials in R[x] or C[x]
(c) Homework problems: #14, 18, 19; Daily Update #11, 12
(d) Additional practice problems: #8, 16
Chapter 17. (17A - 17B)
(a) Concepts: Congruence modulo a polynomial, a complete set of representatives (CSR)
modulo a polynomial, the residue of least degree of a polynomial modulo another, the
set of congruence classes modulo a polynomial (see also 23B), the Chinese Remainder
Theorem (CRT) for polynomials
(b) Goals: Determine whether two polynomials are congruent modulo p ∈ F [x] (F a field),
find a CSR modulo p, verify whether a set is a CSR modulo p, determine the number of
elements in any CSR modulo p, find the residue of least degree of a polynomial modulo
p, determine the units and zero divisors of congruence classes modulo p (see also 23B),
write addition/multiplication tables for congruence classes modulo p (see also 23B), solve
equations modulo p, find all solutions to a system of congruence equations modulo p using
the CRT, find a polynomial with bounded degree and with certain points using the CRT
(c) Homework problems: #2, 3, 5, 7, 9, 10, 13, 14, 24, 25
(d) Additional practice problems: #1, 4, 6, 8, 11, 12, 19
Chapter 23. (23B)
(a) Concepts: The commutative ring F [x]/hpi consisting of congruence classes modulo p ∈
F [x] (where F is a field)
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Math 558: Introductory Modern Algebra, Fall 2015, Fall 2015
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(b) Goals: Determine the number of elements of F [x]/hpi, find all units and all zero divisors
of F [x]/hpi
(c) Homework problems: #10
(d) Additional practice problems: #9, 11
Proof-based problems. Here is a collection of additional (some more challenging) proofs to
practice with that require knowledge from our course. Make sure to study the homework
problems from each section as well! Note that several of these problem appear in the
book, the Daily Update, or homework!
1. Let p be a prime number, and let a, b be integers. Without referring to Unique Factorization, prove the Property of Primes: if p divides ab, then p must divide a or p must divide
b. [Hint: Prove the following equivalent statement: If p - a and p - b, then p - ab.]
2. Without the use of a calculator, show that 74 ≡ 1 mod 10, and use this to find the least
positive residue of 12345678910 mod 10. Carefully show each of your logical steps.
3. Show that out of any three consecutive integers, at least one must be divisible by 3.
4. Show that if a, b and c are three consecutive integers, then a3 + b3 + c3 is not a perfect
square unless it is a multiple of three. That is, if a3 + b3 + c3 = d2 for an integer d, then
d must be a multiple of three. [Hint: Consider the equation modulo three, and apply
Fermat’s Little Theorem.]
5. Consider natural numbers a, b > 0. Show that a | b if and only if a7 | b7 .
q
√
6. Prove that 35 is irrational. Prove that 3 20 is irrational.
7. Let a, b and c be integers. Show that if (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.
[Hint: This problem can be solved using Bézout’s Theorem or the Fundamental Theorem
of Arithmetic.]
8. Prove that if R is a finite commutative ring with no zero divisors, then it must be a field.
9. If R and S are rings, describe the units of R × S.
√
√
√
10. Let Q( 3) denote the set {a + b 3 : a, b ∈ Q} ⊆ R. Show that Q( 3) is a field.
a b
11. Let S =
: a, b ∈ R . Show that the sum (and product) of two elements of S
−b a
stays in S. Then show that S is a commutative ring.
12. If R is a commutative ring with no zero divisors, prove that for all f (x), g(x) ∈ R[x],
deg(f g) = deg(f ) + deg(g).
13. Find a counterexample to (12) when R has zero divisors.
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14. Let R and S be rings, and let ϕ : R → S be a ring homomorphism. Show that if r is a
unit in R, then ϕ(r) is a unit in S.
15. If R is a commutative ring with no zero divisors, show that the units of R[x] are exactly
the units of R.
16. If F is an infinite field, show that f (α) = 0 for every α ∈ F if and only if f (x) is the zero
polynomial. [Hint: Use the Root Theorem]
17. Find a counterexample to (13) when F is finite.
18. Let f (x) = x4 + 3x3 − 6x2 + 12x − 40 ∈ R[x]. Show that 2i is a complex root of f (x).
Then factor f (x) into a product of irreducible polynomials in R[x]. Show your work, and
explain why the polynomials appearing in the factorization are irreducible in R[x].
Computational problems. These problems require the knowledge of some of the topics that
frequently come up in computational problems; a problem on the final exam may require some
combination of these techniques. Many of the problems below are (or are very similar to)
assigned homework problems.
1. Division Algorithm, Euclidean Algorithm, Bézout’s Theorem, and linear Diophantine
Equations (for Z and for F [x]): 3.5, 3.20, 3.57, 14.14, 14.30
2. Unique Factorization (traditional, and for F [x]): 4.10, 4.16, 4.37, 14.42, 14.43
3. Fermat’s Little Theorem: 9.25. Also, can you combine this with CRT to solve a problem
like 12.44?
4. Chinese Remainder Theorem (traditional, and for F [x]): 12.34, 17.19. Could you solve
a problem like 17.21, with three congruences? Can you use CRT to solve 14.10?
5. Arithmetic in fields F and in Z/mZ: 6.24, 6.43, 6.57.
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