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).
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
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