Math 302 Exam 1 Solutions
Modern Algebra
Spring 2009
1. (16 points) Prove Theorem 1.10 for positive integers: Every positive integer greater than 1 is the product of primes. This has symbolic form (n > 1) [n is a product of primes]. Note that a product of
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 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 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 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 #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 #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
4/8/2016
StudyBlue Flashcard Printing of Les 305
Declaration of
Independence
the document that declared the
American colonies
independence from England
federal government/Articles of
Confederation
in 1778, the Continental
Congress formed? and
adopted the?
Math 302 Spring 2017
Name
Quiz #1
1. Give an example of a set of integers that does not satisfy the hypotheses of the Well-Ordering
Axiom, but that does satisfy the hypotheses of the generalized Well-Ordering Axiom.
Math 302 Spring 2016 Homework #8 - Solutions
Ch. 3.2 #17: You may have noticed that this was a problem you just got back as part of HW #7.
My bad; I won't count it this time!
Ch. 3.3 #9: If f: Z Z is an isomorphism, prove that f is the identity map. [Hint
Math 302 Spring 2016 Homework #6 - Solutions
Ch. 3.1 #1: a. S = cfw_all odd integers and 0. If we run down the list of axioms, the failure occurs
quickly: closure under addition fails any time we add two non-zero elements of S (that are not
opposites), si
Math 302 Spring 2016 Homework #4 - Solutions
Ch. 2.1 #1(8): Show that a p-1 1 (mod p) for a: a = 2, p = 5. a p-1 = 24 = 16 1(mod 5).
b: a = 4, p = 7. a p-1 = 46 = 4096 1(mod 7), since 4096 = 585(7) + 1. c: a = 3, p = 11.
a p-1 = 310 = 59,049 1(mod 11), si
Math 302 Spring 2016 Homework #9 - Solutions
Ch. 4.1 #6: Which of the following subsets of R[x] are subrings of R[x]? Justify your answer.
Recall when discussing polynomial rings R[x], we always assume that the ring R has an identity
1R. This means that w
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 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 #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 #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 #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 #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 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 2016 Homework #7 - Solutions
Ch. 3.1 #16: 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 log b . a. Is L a ring under these operations? Since the product
of two positive r
Math 302 Fall 2014 Homework #10 - Solutions
Ch. 4.3 #1: b. Find a monic associate of 3x5 - 4x 2 + 1 in Z5[x]. We find the inverse of 3 in Z5:
3 x 2 = 1, so if we multiply our polynomial by 2, the result will be monic: 6x 5 - 8x 2 + 2 =
x 5 - 3x 2 + 2.
Ch.
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 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 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 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 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 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 Spring 2016 Homework #3 - Solutions
Ch. 1.3 #9: Let p be an integer other than 0, 1. Prove that p is prime if and only if it has this
property: Whenever r and s are integer such that p = rs, then r = 1 or s = 1. This is an if and
only if proof, s
Math 302 Spring 2016 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 2016 Homework #5 - Solutions
Ch. 2.1 #16: If [a] = [1] in Zn, prove that (a, n) = 1. Lets rewrite our hypothesis in more familiar
language: [a] = [1] in Zn means a 1 (mod n), or a 1 = nk, for some integer k. Now we can see
that if d is a c
Math 302 Spring 2016 Homework #1 - Solutions
Ch. 1.1 #7(5): Prove that the square of any integer a is either of the form 3k or of the form 3k + 1
for some integer k. Use the Hint (always use the hints thats what theyre there for!). Since a
is an integer i