Math 302 Spring 2007 Homework #6 - Solutions
Ch. 3.1 #2: a. First we need to check that R is a ring. The tables show that R is closed under
both addition and multiplication (no element not in R shows up in the table). In order to
determine if addition in
Math 302 Sp07 Homework #13 Solutions
Chapter 7.1 #1, 5, 6: These were posted last week in the solutions to Homework #12.
Chapter 7.1 #17: Our set is the nonzero real numbers with operation a * b = | a | b. If a and b are
both nonzero reals, then a * b = |
Math 302 Fall 2006 Homework #7 - Solutions
Ch. 3.1 #22: Let L be the set of positive real numbers. Define a new addition and multiplication
on L by a b =ab , and a b = a logb . a. Is L a ring under these operations? Since the product
of two positive real
Math 302 Modern Algebra Homework Solutions 3.1 DEFINITIONS AND EXAMPLES OF RINGS
Spring 2009
2. Let R = cfw_0, e, b, c with addition and multiplication dened by the following tables. It can be shown that R is a ring with this addition and multiplication.
Math 302 Spring 2007 Homework #11 Solutions
Ch. 4.2 #9: If f(x) is relatively prime to 0F, what can be said about f(x)? Since every polynomial
divides 0F, if 1F is the only monic polynomial that divides both, then 1F is the only monic
polynomial to divide
Math 302 Spring 2007 Homework #5 - Solutions
Ch. 2.1 #21: Prove or disprove: if [a] = [b] in Zn, then (a, n) = (b, n) in Z. This turns out to be
true. Let c = (a, n), and d = (b, n). Since [a] = [b], we can write a = b + nk for some integer k.
Since c div
Math 302 Spring 2007 Homework #1 - Solutions
Ch. 1.1 #2: Perform the division on your calculator. Now find the largest integer smaller than or
equal to your display. (If the display is a positive number, just drop the portion after the decimal
point. If t
Math 302 Spring 2007 Homework #0 - Solutions
In this first set of solutions, I am taking extra care to explain what I am doing. This makes the
solutions much longer than they need be. To give you an idea of how little you need write, I will
enclose the ne
Math 302 Homework Solutions 3.2 BASIC PROPERTIES OF RINGS
Modern Algebra
Spring 2009
3. Let R be a ring. (a) Prove that the additive identity (or zero element) is unique. Proof: Suppose 0R and zR are both zero elements for the ring R. We need to show that
Math 302 Spring 2007 Homework #2 - Solutions
Ch. 1.2 #1: e. Find (306, 657) 657 306 = 2 R 45, so (306, 657) = (306, 45). 306 45 = 6 R 36,
so (306, 657) = (45, 36). 45 36 = 1 R 9, so (306, 657) = (36, 9). 36 9 = 4 R 0. The last
nonzero remainder was 9, so
Math 302 Spring 2007 Homework #4 - Solutions
Ch. 2.1 #1: If a b (mod n) and k | n, is it true that a b (mod k)? If a b (mod n), that says
that (a -b) = nx for some integer x. If k | n, then n = ky, for some y Z. So now a - b = nx =
(ky)x = k(yx), and k |
Math 302 Spring 2007 Homework #3 - Solutions
Ch. 1.3 #2: Let p be an integer other than 0, 1. Prove that p is prime if and only if for each a in
Z, either (a,p) = 1 or p | a. First lets write out the logical structure of this statement: Let
P = p is prime
Math 302 Fall 2006 Homework #9 - Solutions
Ch. 1.2 #13: If k = abc + 1, then prove that (k, a) = (k, b) = (k, c) =1. Suppose (k, a) = d. Then d
is a positive integer which divides both k and abc (since it divides a). But then by the two out of
three rule,
Math 302 Spring 2007 Homework #10 - Solutions
Ch. 4.1 #3: (a) List all polynomials of degree 3 in Z2[x]. Now this sounds like a really gross
problem, until you remember that Z2 has only two elements, 0 and 1. A third degree polynomial
must start with x3,
Math 302 Spring 2007 Homework #8 - Solutions
Ch. 3.2 #6: Let R be a ring and b a fixed element of R. Let T = cfw_rb | r R. Prove that T is a
subring of R. The trick in this problem is knowing how to write down two elements of T. If r1
and r2 are two arbit
Math 302 Fall 2006 Homework #13 Solutions
Chapter 4.5 #1: a. By the Rational Root Test, the only possible roots of
are 1 and 2. We find that x = -1 is a root, and after dividing, that
=
. We test the cubic (using the same list of candidates), and see that
Math 302 Exam 3 Solutions
Modern Algebra
Spring 2009
1. (10 points) Prove Corollary 4.30: Every polynomial f (x) of odd degree in R[x] has a root in R. Proof: Suppose f (x) is an arbitrary polynomial in R[x] with odd degree. We need to show that f (x) has
Math 302 Exam 2 Theorems Chapter 3 Rings 3.2 BASIC PROPERTIES OF RINGS
Abstract Algebra
Spring 2009
THEOREM 3.3 For any element a in a ring R, the equation a + x = 0R has a unique solution. That is, for each a R, there exists a unique n R such that a + n
Math 302 Modern Algebra Homework Solutions 3.3 ISOMORPHISMS AND HOMOMORPHISMS
Spring 2009
1. Write out the addition and multiplication tables for Z6 and for Z2 Z3 . Use them to show that Z6 Z2 Z3 . = The addition and multiplication tables for Z6 are + 0 1
Math 302 Modern Algebra Homework Solutions 4.1 POLYNOMIAL ARITHMETIC AND THE DIVISION ALGORITHM
Spring 2009
5. Find polynomials q (x) and r(x) such that f (x) = g (x)q (x) + r(x) and r(x) = 0 or deg r(x) < deg g (x). (a) f (x) = 3x4 2x3 + 6x2 x + 2 and g
Math 302 Homework Solutions 4.2 DIVISIBILITY IN F [x]
Modern Algebra
Spring 2009
4. (a) Prove: For all f (x), g (x) F [x], if f (x)|g (x) and g (x)|f (x), then there exists c F , c 6= 0F such that f (x) = cg (x). This has symbolic form f (x) F [x] g (x) F
Math 302 Modern Algebra Homework Solutions 4.3 IRREDUCIBLES AND UNIQUE FACTORIZATION
Spring 2009
6. Show that x2 + 1 is irreducible in Q[x]. Proof: Use a proof by contradiction. Suppose x2 + 1 is reducible in Q[x]. We then need to obtain a contradiction.
Math 302 Modern Algebra Homework Solutions 4.4 POLYNOMIAL FUNCTIONS, ROOTS, AND IRREDUCIBILITY
Spring 2009
3. Use the Remainder Theorem to determine if h(x) is a factor of f (x). (b) h(x) = x 1 and f (x) = 2x4 + x3 + x 3 in Q[x] 2 4 According to the Remai
Math 302 Homework Solutions 4.5 IRREDUCIBILITY IN Q[x]
Modern Algebra
Spring 2009
1. Use the Rational Root Test to write each polynomial as a product of irreducible polynomials in Q[x]. (e) 2x4 + 7x3 + 5x2 + 7x + 3 By the Rational Root Test, if r/s is a r
Math 302 Homework Assignments Assignment 7 due Monday, April 6 4.3
Modern Algebra
Spring 2009
Irreducibles and Unique Factorization
6. Show that x2 + 1 is irreducible in Q[x]. [Hint: Use a proof by contradiction. Suppose x2 + 1 = (ax + b)(cx + d) for some
Math 302 Modern Algebra Homework Solutions 7.5 CONGRUENCE AND LAGRANGES THEOREM
Spring 2009
1. Let K be a subgroup of a group G. Prove: For all a G, K a = K if and only if a K . Proof: Suppose a is an arbitrary element of G. We need to prove K a = K if an
Math 302 Exam 1 Information
Modern Algebra
Spring 2009
Date: Friday, February 27 Covering: Chapter 1, Sections 1, 2, 3, 4 Chapter 2, Sections 1, 2, 3 General Comments You will need to bring a large (8 1 11) blue book to the exam. You should also bring the