## Navigation

- Introduction.
**Chapter 1:**Section 1.3, Section 1.4, Section 1.5.**Chapter 3:**Section 3.1, Section 3.2, Section 3.3, Section 3.5, Section 3.6, Section 3.7.**Chapter 2:**Section 2.2, Section 2.3, Section 2.4.

Back to the Main Page.

## Inroduction

### Reading and Assignments

*Carefully*read our Course Web Site.- Subscribe to receive notifications from Piazza and Canvas, and set forwarding for your preferred e-mail. (See here.)
- Create an account at CoCalc. You should have received an invitation to collaborate in the "Math 506 - Summer 2017" project that I've created. You don't need to worry much about it for now, but watch the videos I've posted to see how it works.
- Sign in or create an account for our course in Piazza. (See here.) You should have received an e-mail about your enrollment by the first day of class.
*Quickly*check sections 1.1, 1.2 and 2.1 from the book. (You can leave 2.1 for later, perhaps.) You basically have to remember the basics of: complex numbers (Section 1.2), matrices (see, for instance, the Wikipedia entry), induction (see Section 1.1), binomial theorem (see Section 1.1) and functions (including one-to-one, onto and invertibles functions -- Section 2.1).*Don't spend too much time here!*You can always come back as you need it.- On induction, you can check this lecture from my 504 course. In particular, you can watch some videos with examples.
- Write down all questions about the above topics to bring to our (online) lecture. You can also type them, ahead of time, in the file "Questions.tex" (a LaTeX file) in CoCalc (under the "Math 506 - Summer 2017" project). You might see others questions in there (everyone will use the same file).
- Check if you are prepared for our lecture with Zoom:
- Check LiveOnline@UT, the official page for Collaborate.
- Perform a System Check to see if things should work with your computer.
- Check System Requirements.
- If available, do a Test Flight, as explained in an Announcement on Canvas.
- Check the Resources for Participants.

### Videos

- Introduction to
CoCalc: here I show
*a little*about LaTeX and the use of CoCalc. **Optional**: You can watch these videos on What is Algebra?, where I give a brief answer to the question, and on Algebraic Structures (made for Math 457), which goes over a few topics we will cover, but many others that we won't. I think these make a good introduction to our course.

### Problems

**None.**

### Outcomes

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

- understand the course structure;
- be familiar with Canvas, CoCalc and Piazza;
- have a (very) basic understanding of LaTeX;
- you should (from the reading):
- know how to compute with complex numbers and matrices.
- know basic concepts of functions, such as one-to-one, onto, invertible and inverses.
- know the formula (and how to use it) from the
*Binomial Theorem:*\[ (a+b)^n = \sum_{i=0}^n \binom{n}{i} a^i b^{n-i}. \] - know how to do simple proofs with induction.

## Section 1.3

### Readings and Assignments

- Division algorithm: review how to do it by hand and careful with negative numbers! (Remainder is always non-negative!)
- Definition of prime and composite numbers.
- Skip algorithm on top of pg 39.
- Definition of divisibility and GCD.
*Euclid's Lemma*(Theorem 1.38).- Skip Proposition 1.42.
- $\sqrt{2}$ is irrational.
*Extended Euclidean Algorithm*.- Skip Proposition 1.46.
- $b$-adic digits (representation on base $b$) and applications on divisibility criteria.

### Videos

- Long division with negatives.
- Proof of the
following
**Basic Lemma:**Suppose that $d \mid a$. Then $d \mid (a+b)$ iff $d \mid b$. - Euclid's Lemma.
- Example of the Extended Euclidean Algorithm.
- \(b\)-adic representation (1.53 with \(b=4\)).
- Problems: 1.46(ix), (x), 1.60.
- A few words about Example 1.49.
**Optional**: Computations in Sage. (Sage is what Coclac uses for computations, hence the previous name, "Sage Math Cloud".)

### Problems

**Section 1.3:** 1.46(i) to (x), 1.50, 1.52, 1.53,
1.55, 1.57.

### Outcomes

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

- be able to perform long divisions (even with negatives);
- understand divisibility (Problem 1.46(i) to (v));
- be able to write simple proofs about divisibility (such as Problem 1.50);
- know what are prime and composite numbers and be able to distinguish them (for small numbers);
- know what the GCD is and how to compute it for small numbers (Problem 1.46(vi) and (vii));
- know how to compute GCD of large numbers and how to write it
as linear combination of the given numbers (
*Extended Euclidean Algorithm*or*Bezout's Theorem*- Problem 1.55); - be able to use the fact that the GCD is a linear combination of the numbers in simple proofs (Problem 1.57);
- know how to express numbers in different bases (Problem 1.53) and use it (Problem 1.52).

## Section 1.4

### Reading

*Fundamental Theorem of Arithmetic*(Theorem 1.51 and its corollaries).- Divisibility (Lemma 1.54), GCD and LCM (Proposition 1.55) using prime factorization.
- $\operatorname{lcm}(a,b) \cdot \gcd(a,b) =\left| a \cdot b \right|$ (Proposition 1.56).

### Videos

### Problems

**Section 1.4:** 1.68, 1.69, 1.70(i), 1.76(ii).

### Outcomes

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

- know how to factor integers into a product of powers of primes and check divisibility and compute GCDs (Problem 1.69) and LCMs (Problem 1.76(ii)) using these factorizations;
- apply this concept to prove basics properties (Problem 1.70(i));

## Section 1.5

### Reading

- Congruences (Proposition 1.58 and Corollary 1.59 are important).
- Application to rule out number which are not perfect squares (Example 1.61).
- Skip Proposition 1.62 (although it is very nice!).
- Fermat's Theorem (Theorem 1.64).
- Divisibility Criteria (like Corollary 1.65).
- Congruences with powers (Corollary 1.67 and Example 1.68).
- Solving linear congruences (Theorem 1.69, Examples 1.71 and 1.72).
*Chinese Remainder Theorem*(Theorem 1.73, Examples 1.74 and 1.75, as well as Proposition 1.76 and Example 1.77).

### Videos

- General remarks on congruences.
- Powers in congruences.
- Corollary 1.65 (divisibility criteria for 3 and 9).
- Problems 1.77(vi), (vii), 1.78(ii) (and extra example), 1.91(i).
- CRT with non-relatively prime moduli.
**Optional**: Computations with Sage. (Sage is what CoCalc uses for computations, hence the previous name "Sage Math Cloud".)

### Problems

**Section 1.5:** 1.77, 1.78, 1.79, 1.80, 1.81,
1.83, 1.85, 1.86, 1.91, 1.95.

### Outcomes

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

- understand when numbers are congruent;
- be able to reduce numbers module $m$;
- be able to apply congruence to solve problems (Problems 1.79 and 1.85);
- be able to reduce large powers modulo $m$ (Problem 1.81);
- be able to solve linear congruence equations (Problem 1.78);
- be able to solve systems of congruences (
*Chinese Remainder Theorem*- Problems 1.91 and 1.95).

## Section 3.1

### Reading

- Definition and basic properties of rings.
- Skip Examples 3.10 and 3.11.
- Subtraction and multiplication by integers.
- Integral Domains.
- Subrings.
- Integers modulo $m$.
- Units and divisbility.

### Videos

- General Remarks on rings.
- Examples of rings.
- Other operations (subtraction, multiplication by integers, powers).
- Integers modulo \(m\).
- Integral domains (and zero divisors).
- Subrings (including Gaussian integers, \(\mathbb{Q}[\sqrt{2}]\)).
- Units.
- Problem 3.2.
**Optional**: Computations with Rings in Sage. (Again, Sage is what CoCalc uses for computations.)

### Problems

**Section 3.1:** 3.1 except (v) and (viii), 3.2, 3.3, 3.6,
3.8(i), 3.13, 3.15(i), (ii).

### Outcomes

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

- know the definition, basic properties and examples of rings;
- be able to prove basic properties of abstract rings (Problems 3.2, 3.3);
- understand the difference between the multiplication of two elements of a ring and the multiplication of one element of a ring times an integer;
- know the definition and basic properties of an integral domain (Problems 3.8(i) and 3.15(i) and (ii));
- know the definition of a subring and know how to determine if a subset of a ring is a subring (Problem 3.13 and 3.15(i) and (ii));
- understand the rings $\mathbb{Q}[\sqrt{2}]$ (a subring of $\mathbb{R}$) and $\mathbb{Z}[\mathrm{i}]$ (a subring of $\mathbb{C}$);
- understand the ring of
*integers modulo $m$*, denoted by $\mathbb{I}_m$ in the text, and know how to perform computations in this ring (Problem 3.6); - know the definition of units and know how to find them (Problem 3.6);
- understand the generalization of divisibility on the context of abstract rings.

## Section 3.2

### Reading

- Definition and basic properties of fields.
*Skim*over construction of the fraction field.- Subfields.
- Prime fields.

### Videos

- Fraction field.
- Problems 3.17(iv), (vii), 3.19, 3.27(i).

### Problems

**Section 3.2:** 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).

### 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)).

## Section 3.3

### Reading

- Watch this video about
Section 3.3 (
*before reading the section!*). - 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.

### Videos

### Problems

**Section 3.3:** 3.29, except (i), 3.30, 3.32, 3.37 (this
one should be after 3.5).

### Outcomes

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

- 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 of this section are of this sort).

## Section 3.5

### Reading

- 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.

### Videos

- Example of Long Division of Polynomials.
- Example of extended Euclidean algorithm.
- Problems 3.56(vi), (vii), 3.64.

### Problems

**Section 3.5:** 3.56 from (i) to (vii) (in (vii) it
should say $k=\mathbb{F}_p$, not $k= \mathbb{F}_p(x)$), 3.58, 3.62,
3.64.

### 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.

## Section 3.6

### Reading

Read Theorem 3.84 and Example 3.85 *only*!

### Videos

None.

### Problems

**Section 3.6:** None.

### Outcomes

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

- 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.

## Section 3.7

### Reading

- 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).

### Videos

- 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).
**Optional**: Computations with polynomials in Sage. (Again, Sage is what CoCalc uses for computations.)

### Problems

**Section 3.7:** 3.86, 3.87 except (vii).

### Outcomes

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

- 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).

## Section 2.2

### Reading

- Focus on
*computations*! - Permutations (matrix notation, computing composition and
inverses, inverse of a composition).
**Careful:**Like functions, read compositions from right to left! - Cycles.
- Decomposition into disjoint cycles/complete factorization (Properties 2.24, Theorem 2.26).
- Conjugation of permutations (Lemma 2.31 and Proposition 2.32).
- Decomposition as composition of transpositions (Proposition 2.35)
- You may skip Example 2.36 and Lemmas 2.37 and 2.38. (They are important to the proof of the formulas for sign below.)
- Sign (Definition, Theorems 2.39 and 2.40 and Corollary 2.41).
- Skip Example 2.42.

### Videos

### Problems

**Section 2.2:** 2.21 ((ii) is easier after 2.3), 2.22,
2.25, 2.26, 2.34, this question
on permutations.

### Outcomes

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

- know the definition and basic properties of permutations;
- know that permutations in general do not commute,
but
*disjoint*cycles do; - be able to compute compositions, inverses, factorization into disjoint cycles, factorization into transpositions and sign of permutations (Problem 2.22 and extra problem);
- understand how conjugation (the
*conjugation*of $\alpha$ by $\beta$ is $\beta \alpha \beta^{-1}$) affects cycles (and products of cycles) and be able to apply this idea.

## Section 2.3

### Reading

- Definition and basic properties of Groups.
- Examples (Example 2.47, except (v), (vii) and (viii), and 2.48, except (vi)).
- Associativity and parentheses (Theorem 2.49).
- Powers/multiplication by integers.
- Order of elements (Definition and Propositions 2.54 and 2.55).
- Skip from Symmetry (pg. 137) to the end of the section.

### Videos

- Remarks on abstract groups and examples.
- Powers and multiplicative/additive notations.
- Order of elements.
- Problems 2.36(v), 2.40.

### Problems

**Section 2.3:** 2.36 (i) to (v) and (viii) to (ix), 2.37,
2.38, 2.40.

### Outcomes

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

- know the definition, basic properties and examples of groups;
- know how to prove basic properties of groups (Problems 2.37 and 2.40.);
- know how to translate from multiplicative notation to additive notation;
- know how to compute order of elements (Problem 2.38).

## Section 2.4

### Reading

- Definition of subgroups.
- Subgroup criteria (Propositions 2.68 and 2.69).
- Alternating subgroups \(A_n\).
- Cyclic subgroups.
- Skip from Proposition 2.71 to 2.73.
- Order of a group.
- Skip from Proposition 2.75 to Lemma 2.82.
- Lagrange's Theorem (Theorem 2.83 -- you can skip the proof) and its corollaries (from 2.85 to 2.87, you can skip 2.84).

### Videos

### Problems

**Section 2.4:** 2.52 (i) to (v) and (x) to (xi), 2.54,
2.55, 2.57.

### Outcomes

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

- know how to check if a subset of a group is a subgroup (Problem 2.54);
- know the definition of the alternating group $A_n$;
- know the definition of cyclic groups and how to check if a group is cyclic (Problem 2.52(x) and (xi));
- know
*Lagrange's Theorem*and how to apply it in simple proofs (Problem 2.57).