Polynomial rings:
The most relevant page of the Definitions and Theorems
file is page 8
Monday, June 1: Polynomials with coefficients in a Field (or a ring) as a ring
🎥
Video: You can have polynomials with
coefficients in a ring (like real numbers or mod numbers), and the set
of polynomials is a ring
🎥
Video: Fields are
particularly nice rings. Mostly we study polynomials where the
coefficients are in a Field
🎥
Video: In a F[x] which is the ring of
polynomials with coefficients in a field, you can do division with
remainders
🎥
Video: Examples of polynomial division with mod number coefficients--don't skip me!
🎥
Video: Me trying to explain the
assignment
🎥
Video: Greatest common divisors of two
polynomials (no homework problem--your assignment is to watch and
appreciate!
✎ Assignment:
4.1 # 1 a, b, c, d, 5 a, b, c, d and 6* a, b, c, d, e * In number
6, decide whether each subset is a ring or not by using examples to
decide if the set is closed under addition, (additive inverses) and
multiplication. You do not need to give a proof.
Tuesday, June 2: Irreducibility and factorization of polynomials
🎥 Video: Introduction to
irreducibility
🎥 Video: Irreducibility and
the Quintic Problem
🎥 Video: The Factor Theorem--use this technique also for finding remainders.
🎥 Video:
Numberphile explaining the Fundamental Theorem of Algebra
🎥 Video: Factoring and
irreducibility over the rational, real and complex numbers
🎥 Video: Factoring and irreducibility over
mod number fields (prime mods)
🎥 Video: The remainder theorem (useful for 4.4 # 2, 3)
✎
Assignment:
1. Prove (write the proof of) the factor theorem (Theorem 59). For full credit, do not use the remainder theorem.
4.3 # 10 a, b, 11, 12
4.4 # 2 a, b, c, d, 3 a, b, c, d, 8a, b, c, d, e, f
Note: The April 17 discussion
session had some great examples in the Q&A including how to do
4.4#2, and how to get started on 4.1 #6. That video is up in
Canvas--go check it out!
Wednesday, June 3: Study for the polynomial rings quiz
The quiz will consist of:
1. A question that is one of the following for a polynomial f(x) with coefficients in a field.
- Prove that if f(a)=0 then (x-a) is a factor of f(x)
- Prove that if (x-a) is a factor of f(x) then f(a)=0
- Prove that if g(x) is a degree 1 polynomial, with remainder r(x), then r(x) is a constant, and g(a)=0 then f(a)=r(x)
2. A division problem like one of 4.1 5a-d
3. A subring problem where you explain using examples, like 4.1 6a-6e
4. A problem where you find the factorization of a polynomial over a field, possibly similar to 4.3 # 12
5. A problem where you determine if a polynomial is irreducible similar to 4.3 # 10, 11 or 4.4 # 8 (4 polynomials)
Thursday, June 4: Take the polynomial rings quiz.