Lecture 7

Navigation

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

Before the Lecture

• Read Section 3.2:
  • Definition and basic properties of fields.
  • Skim over construction of the fraction field.
  • Subfields.
  • Prime fields.
• Watch the videos related to this section (after reading it):
  • Fraction field.
  • Problems 3.17(iv), (vii), 3.19, 3.27(i).
• Watch this video about Section 3.3 (before reading the section!).
• Read Section 3.3:
  • Read it all, mentally converting from sequences to polynomials. May skip power series.
  • Definitions.
  • Watch for Lemma 3.24: property of degrees for non-domains.
• Watch the videos related to this section (after reading it):
  • Word about the derivative.
• 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 06/23 by 11:59pm.

Section 3.2:	Turn in: 3.17(ii), (iii), 3.23, 3.27(ii).
	Extra Problems: 3.17, 3.19, 3.20 (hint: in a domain, if $a \neq 0$ and $ax=ay$, then $x=y$; use that to show that if $a \neq 0$ and $R$ is a finite domain, then $\{ ax \; : \; x \in R\} = R$; use that to show $a$ is a unit), 3.23, 3.27(i), (ii).

 

Section 3.3:	Turn in: 3.29(ii), 3.30.
	Extra Problems: 3.29, except (i), 3.30, 3.32, 3.37 (this one should be after 3.5).

 

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:

• know the definition, basic properties and examples of fields;
• be able to recognize which rings are fields (Problem 3.23(i));
• have an idea about the construction of fields of fractions (Problem 3.17(vii));
• know how to determine if a subset of a field is a subfield (Problems 3.17(v) and (vi) and 3.23);
• understand and be able to recognize prime fields (Problem 3.27(ii));
• know the definition and basic properties of polynomial rings, including degrees;
• know how to compute with polynomial rings;
• apply previous concepts for abstract rings to polynomials (units, integral domain, divisibility, etc. -- most problems are of this section are of this sort).