Lecture 8

Navigation

• Back to Main Page.
• Back to the Lectures.
• Before the Lecture
• Related Problems
• In Class
• Outcomes

Before the Lecture

• Read Section 3.5:
  • The important topics: long division, GCD, Euclidean algorithm. Make sure you know this and read the rest quickly.
  • Division Algorithm (Theorem 3.46).
  • Skip Proposition 3.47.
  • Divisibility by \(x-a\) (Proposition 3.49).
  • Degree and number of roots (Theorem 3.50 and Examples 3.51).
  • Skip from Corollary 3.52 to Theorem 3.55.
  • GCD and its basic properties.
  • Skip Theorem 3.59 to Proposition 3.63.
  • Irreducible (or prime) polynomials and their basic properties.
  • Euclidean Algorithm (Theorem 3.71 and Examples 3.72 to 3.75).
  • Skip from Euclidean Rings (on pg. 267) to the end of the section.
• Watch the videos related to this section (after reading it):
  • Example of Long Division of Polynomials.
  • Example of extended Euclidean algorithm.
  • Problems 3.56(vi), (vii), 3.64.
• Read Section 3.6: read Theorem 3.84 and Example 3.85 only!
• Read Section 3.7:
  • Rational roots (Theorem 3.90).
  • Gauss's Lemma (Lemma 3.92).
  • Factorization into content and primimitve (Lemma 3.93 to Theorem 3.96).
  • Checking reduction modulo \(p\) (Theorem 3.97).
  • Eisenstein Criterion (Theorem 3.102).
• Watch the videos related to this section (after reading it):
  • Irreducible polynomials of degree less than or equal to \(3\).
  • Example of factorization into content times primitive.
  • Problems 3.86 (xiv), 3.87(iii), (vi), (x).
  • Computations with polynomials in Sage.
• Write down all questions about the above topics to bring to our (online) lecture. (You can also type them in the file "Questions.tex" in SageMathCloud.) Comments about the videos are welcome!
• Work on the assigned problems for these sections. (See Related Problems below.) You don't need to finish them, but try to work on as many as you can and the bring your questions to class.

 

Related Problems

The "turn in" problems are due on 06/26 by 11:59pm.

Section 3.5:	Turn in: 3.56(v), 3.58, 3.62.
	Extra Problems: 3.56 from (i) to (vii), 3.58, 3.62, 3.64.

 

Section 3.7:	Turn in: 3.86 (vi), 3.87 (v), (ix).
	Extra Problems: 3.86, 3.87 except (vii).

 

In Class

In class:

• We will discuss the reading and pace.
• I will discuss the main points.
• I will answer questions about the sections covered.
• I will answer any other questions.
• We can work on the HW problems.

 

Outcomes

After the assignment (reading and videos before class) and class, you should:

• see how integers and polynomials over a field (i.e., with coefficients in a field) are very similar;
• know how to prove basic properties of polynomials over a field (using the same ideas as we've used for integers, like divisibility, GCD, the fact the GCD is a linear combination, etc.) (Problems 3.62 and 3.64);
• know how to compute long division and GCD of polynomials (Problem 3.58);
• know, and be able to apply, that the remainder of the division of $f(x)$ by $(x-a)$ is $f(a)$;
• know the relation between the degree and the number of roots of a polynomial;
• know the definition and properties of irreducible polynomials and see that they correspond to primes in the context of integers;
• know that polynomials over fields factor "uniquely" (up to a multiplication by constants) as product of powers of irreducibles (the analogue to the Fundamental Theorem of Arithmetic for integers), and apply this to simple proofs and computations of GCD and LCM.
• know how to factor simple polynomials as product of powers of irreducibles;
• know the possible rational roots of a polynomial with integral coefficients (Rational Root Test -- Problem 3.86(ii) to (iv));
• know the relation between irreducible polynomials in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ (Problem 3.86(iv) to (xiv));
• know what a primitive polynomial is (Problem 3.86(v) to (viii));
• know how to compute the content of a polynomial with coefficients in $\mathbb{Q}$ (Problem 3.86(ix) and (x));
• be able to test if an integral polynomial (i.e., with coefficients in $\mathbb{Z}$) is irreducible over $\mathbb{Q}$ by reducing modulo primes (Problem 3.87);
• be able to apply the Eisenstein Criterion to check if an integral polynomial is irreducible over $\mathbb{Q}$ (Problem 3.87).