Marywood University Department of Mathematics Course Information for Math 321 Abstract Algebra Craig M. Johnson

Joseph-Louis Lagrange (1736-1813)

An introduction to Abstract Algebra. This course is required for both mathematics and math/secondary education majors. Topics include groups, rings, fields, and homomorphisms. The textbook is Abstract Algebra by Thomas W. Hungerford and published by Saunders (Harcourt Brace). The course grade is determined by three tests, five or six homework sets, and a final examination. Useful websites are listed at the end of this page. One VERY useful website is http://www.math.niu.edu/~beachy/aaol/frames_index.html which has a nicely organized presentation of key definitions and theorems. Prerequisite: Math 202 (Calculus II); Recommended prerequisite: Math 323 (Number Theory)

Sample Problems for Test #1

• If a is an integer, prove that a² is not congruent to 2 (mod 4) or to 3 (mod 4).
• If a is congruent to b (mod 2n), then prove that a² is congruent to b² (mod 4n).
• Give three examples of equations of the form ax = b in Z₂₀ that have no solution.
• Prove or disprove: Z_n is an integral domain for all positive integers n.
• Define a new addition and multiplication on Z by a # b = a + b - 1 and a * b = ab - (a + b) + 2. Prove that with these new operations Z is an integral domain.
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• Let U be an integral domain.
  • Prove that cancellation is valid in U i.e. If a is not = 0_R and ab = ac in U, then b = c.
  • If e is a non-zero idempotent element (this means e² = e), then e is an identity for U.
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• Let R be a ring with identity. If ab and a are units in R, prove that b is a unit.
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• The center of a ring R is defined to be C(R) = {a in R | ar = ra for all r in R }.
  • Prove that C(R) is a subring of R.
  • If f: R --> S is an isomorphism, prove that f(C(R)) = C(S) i.e. that f(C(R)) is the center of S.
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• Let f: R --> S be a homomorphism of rings. We define the kernel of f to be Ker f = {r in R | f(r) = 0_S }.
  • Show that Ker f is a subring of R.
  • Clearly 0_R is an element of Ker f. Show that if 0_R is the only element of Ker f , (i.e. Ker f = {0_R} ), then f is injective.
• Write out the addition and multiplication tables for the ring Z₂ x Z₄ . Show why there cannot be an isomorphism between Z₂ x Z₄ and Z₈ .
• Prove that Z₁₀ is isomorphic to Z₂ x Z₅.
• Give an example of a subset of M(R) that is isomorphic to R. (You must prove it.) Would this subset have to be a subring?
• Prove: Z_p is a field iff p is prime. (You may assume that we know that Z_p is a commutative ring with identity.)

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Sample Problems for Test #2

CHAPTER 4 Arithmetic in F[x].

• List all the polynomials of degree less than 3 in Z₃[x].
• Which of the following subsets of R[x] are subrings of R[x]? (R a ring.)
  • All polynomials with constant term 0.
  • All polynomials with degree 2.
  • All polynomials in which the odd powers of x have zero coefficients.
• Find the gcd of x⁵ + 1 and x³ + 1 in Q[x].
• Let f(x), g(x), h(x) be in F[x] with f(x) and g(x) relatively prime. If h(x) | f(x), prove that h(x) and g(x) are relatively prime.
• Let R be a commutative ring. If a_n ‚ 0_R and a_o + a₁x + a₂x² + . . . + a_nxⁿ is a zero divisor in R[x], prove that a_n is a zero divisor in R.
• Find the four roots of x² - 1 in Z₈[x]. Why doesn't this contradict the theorem that states that if F is a field, then every nonzero polynomial of degree n in F[x] has at most n roots in F ?
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• If F is a field, show that F[x] is not field. (Hint: You can do this by just showing that any nonconstant polynomial (e.g. x) is not a unit. Use a degree argument.)
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• Determine whether or not the following polynomials are irreducible.
  • x³ + 2x + 1 in Z₅[x]
  • x² - 13 in Q[x]
  • x⁴ - 5 in R[x]
  • x⁵ - 2x⁴ + 3x² + 4x + 1 in Z₇[x]

CHAPTER 5 Congruence in F[x].

• If p(x) has degree k in Z_n[x], how many distinct congruence classes are there modulo p(x)?
• Write out the addition and multiplication tables for Z₂[x]/(x³ + x + 1).
• List the distinct congruence classes of Z₃[x]/(x² + x + 2). Do they form a field? Why?
• Show that Q[x]/(x² + x) is not an integral domain.
• Show that Q[x]/(x² - 2) is a field. Find the inverse to [x].

CHAPTER 6 Ideals and Quotient Rings

• Let R and S be rings and let f : R --> S be a ring homomorphism. Show that Ker f is an ideal in R.
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• Is the set J of all polynomials in Z[x] with odd constant terms an ideal? Why or why not?
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• List the principal ideals of Z₆. If p is prime, show that every (nonzero) ideal in Z_p must be a principal ideal.
• If I is an ideal in a ring R, prove that congruence modulo I is transitive. In other words show that if a = b (mod I) and b = c (mod I), then a = c (mod I). (Here "=" means "is congruent to".)
• Use the First Isomorphism Theorem to prove that Z₂₀/(5) is isomorphic to Z₅ .

   

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Sample Problems for Test #3

• What is the order of each of these groups? i) Z₉ ii) D₇ iii) U₂₀ iv) S₅
• The dihedral group D₃ of consists of the six symmetries of an equilateral triangle. These are the three counterclockwise rotations r₀, r₁, r₂ of 0, 120, and 240 degrees, respectively, of the triangle about its center and the three reflections s, t, and u of the triangle across its altitudes. Show that D₃ is nonabelian.
• Suppose that G is a group with operation *. Define a new operation # on G by a # b = b * a. Show that G is a group with operation #.
• Prove that R* x R is a group under the operation * defined by (a, b)*(c, d) = (ac, bc + d).
• Find the order of each element of the group U₁₂. What is the inverse of 5 in U₁₂?
• If a, b are elements in G, prove that
  • | bab^-1 | = | a |
  • | ab | = | ba |
• Prove that the center Z(G) is a subgroup of G.
• Show that (1, 0) and (0, 2) generate the additive group Z x Z₇.
• Let Z and E be the groups of integers and even integers respectively with the operation of addition. Show that Z is isomorphic to E.
• Let G be a nonabelian group and H an abelian group. Prove that G cannot be isomprophic to H.
• Let G be an abelian group and let a, b be elements of G of finite orders m, n respectively. If (m, n) = 1, show that the order of ab is mn.

Useful Websites

http://www.math.niu.edu/~beachy/aaol/frames_index.html (On-line course in abstract algebra. Contains key definitions and theorems.)

http://www.maa.org (Mathematical Association of America)

http://archives.math.utk.edu (Math Archives)

http://forum.swarthmore.edu (Math Forum)

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update: February 11, 2002

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