DUE DECEMBER 10, 2020
You may refer to your text, class notes, and other written work from this course. Do not confer with anyone else on these problems nor access someone else's solution via the Internet or other means. Clarification questions should be directed to Prof. Buehrle.
Your portfolio should include:
A. Your reflections relating the reading from Mathematics: The Science of Patterns by Keith Devlin to your studies in mathematics, especially Abstract Algebra; how has your study thus far of collegiate-level mathematics expanded or given greater definition to your view of mathematics? The length should be about three pages. (This reading is posted on Joule.)
B. Your solutions to the following problems as well as a brief introduction to each problem relating it to the basic concepts of this course. You may discuss concepts at a high level, assuming that your reader is familiar with the content of the course.
1. Show (Z, *) is an abelian group, given a * b = a + b + 42 for any a, b ∈ Z.
2. Write the following as the product of irreducible polynomials in the specified polynomial rings. In other words, factor the polynomial so that coefficients of the factors are elements of the indicated set.
x^4 - 49 in Z[x], Q[x], R[x], and C[x].
3. Consider S = {[0], [3], [6], [9], [12]} with the addition and multiplication of congruence classes defined for Z15.
(i) Make a Cayley table for each operation.
(ii) This set is a subring of Z15 since it is a subgroup under addition and closed under multiplication. Does it also satisfy the properties of a field? That is, do all non-zero entries have a multiplicative inverse? Justify your response.
4. Let G = {(x, y) | x, y ∈ Z} where the group operation ⊕ is defined by
(a, b) ⊕ (m, n) = (a + m, b + n).
(i) Show that φ : G → Z where φ((x, y)) = x - 3y defines a group homomorphism.
(ii) Find the kernel of this homomorphism, recall that
ker φ = {(x, y) ∈ G | φ((x, y)) = 0}.
Your portfolio should be typed in LaTeX. If you need help typesetting anything contact Prof. Buehrle.
