# Assignment #2

## Transcription

Assignment #2

2015 Spring MATD01 Due: Thurs 5 Mar Assignment #2 INSTRUCTIONS. • Put your legal first / last name and your preferred name and your UofT student number on the front of your submission. Submit your solutions directly to the instructor at the beginning of class on the due date. • You can only submit one assignment (your own). Each person registered for the course must hand in his or her assignment in person. • Show all of your work and give complete arguments, but do not include extraneous information. Do not answer a problem in more than one way. You will be penalized for including non-essential information or multiple solutions to a single problem. • All of the problems on this assignment can be solved using material taught in MATD01 lectures up to and including the lecture on Thursday 12 February. You may be penalized for using material taught after 12 February. (In particular, you are not permitted to use the so-called “advanced tests” for irreducibility. Which tests are simple and which are advanced will be discussed at the beginning of the 24 February lecture.) • If you use a result that was proven in class, write “proven in class” to indicate so. You do not have to re-prove anything that was proven in class. Do not use theorems that were not mentioned in class. (If in doubt, ask the instructor during office hours or by email.) • Only submissions written entirely in ink (or typed up) with no correction tape or correction fluid marks can be considered for re-grading. • LATE ASSIGNMENTS WILL NOT BE ACCEPTED FOR ANY REASON. • WRITE LEGIBLY. In all of the questions below, the addition and multiplication operations associated to each ring are the standard ones unless mentioned otherwise, and multiplication is always commutative. Problem 1. If R is any ring, p(x) ∈ R[x] is a monic polynomial of degree greater than or equal to 1, and I = (p(x)), then prove that R[x]/I is an integral domain if and only if p(x) is irreducible over R. [4 marks] Z[x] a field? You have two options: either use a (x2 + 1) theorem from class to decide that it is a field, or exhibit a nonzero element Z[x] of 2 that is not a unit. [1 mark] (x + 1) Z[x] Problem 2 (b). Show that 2 is isomorphic to a subring of C. [3 marks] (x + 1) Problem 2 (a). Is Problem 2 (c). Taking inspiration from parts (a) and (b), list five different rings that are isomorphic to subrings of C, subject to the following conditions: all are integral domains, none are isomorphic to C itself, and exactly three of them are fields. Explain how you arrived at your list. [6 marks] Problem 3 (a). Let S be a nonempty set with finitely-many elements. Describe the polynomials of degree 1 in P(S)[x] that have roots in P(S). Explain how you arrived at your answer. [3 marks] Problem 3 (b). Let a and b be two distinct elements in S. If p(x) = Sx2 + {a, b} x + {a} and I = (p(x)), then is P(S)[x]/I an integral domain? Explain. [4 marks] Problem 3 (c). Let S be a nonempty set with finitely-many elements. Which monic degree 2 polynomials in P(S)[x] are reducible over P(S)? Explain your work. [4 marks] Problem 3 (d). If S is a nonempty set with finitely-many elements and I / P(S)[x] is a proper ideal generated by a monic polynomial of degree 1 or 2, then when exactly is P(S)[x]/I an integral domain? Explain your work. [3 marks] 2 Problem 3 (e). Recall the following “weak” version of the Fundamental Theorem of Algebra: A polynomial of degree n ≥ 1 in R[x] has at most n roots in R. Does this version of the Fundamental Theorem of Algebra hold when we replace R with P(S), where S is any nonempty finite set? Explain. [2 marks] Problem 4 (a). Identify all of the ideals I /Z2 [x] that contain polynomials of degree 4 and which are maximal and/or prime. Explain your work and be as precise as possible in your identification, by providing generators whenever possible. [6 marks] Problem 4 (b). Identify all of the ideals I / R[x] that contain polynomials of degree 3 and which are such that R[x]/I is not a field. Explain your work and be as precise as possible in your identification, by providing generators whenever possible. (Hint: The Intermediate Value Theorem may be of use.) [4 marks] Total number of marks: 40 3