Math 441 - Abstract Algebra - Schedule

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Much of the development of modern abstract algebra began by studying how to solve algebraic equations, in particular polynomial equations of the form $a_0+a_1x+a_2x^2+a_3x^3+\cdots a_nx^n=0$. If the coefficients are integers, then what can we say about the solutions? What can we say about the solutions if the coefficients are instead rational numbers, real numbers, or complex numbers?

The quadratic formula gives all solutions to $a_0+a_1x+a_2x^2=0$.
In the 1500's, Girolamo Cardano (1501-1576) published a solution to the general cubic equation.
Shortly afterwards, his student Lodovico Ferrari (1522-1565) published a solution to the general quartic equation.

At this point in mathematical history, the next obvious goal was to find the solution to the general quintic equation. This problem remained open into the 1800s, where finally Abel showed that obtaining a formula using radicals was in general impossible. Galois invented group theory to describe the symmetries of solutions to polynomial equations. Most of our modern view of algebra has its roots in the history of solving the general quintic.

A lot of new words have been created since the 1500s to describe different types of number systems. To start this semester, I'd like us to explore the properties of the rationals, and then from them develop the concepts of field, ring, integral domains, and more.

The rational numbers have two group structures. Under addition, the set of rational numbers $(\mathbb{Q},+)$ forms an Abelian group. Under multiplication, we almost have another Abelian group with one problem. The number 0 does not have a multiplicative inverse. We'll let $\mathbb{Q}^*$ denote the set of rational numbers, excluding zero, and then we see that $(\mathbb{Q}^*,\cdot)$ is another Abelian group. When we look at how addition and multiplication interact together, the properties of addition and multiplication satisfy the distributive laws $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$.

Definition (Field)

A field is a set $F$ together with two binary operations $+$ and $\cdot$ that satisfies the following properties:

$(F,+)$ is an Abelian group. Let $F^*$ be the set $F$ with the additive identity removed.
$(F^*,\cdot)$ is an Abelian group.
The distributive laws hold, namely we have $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$.

Problem 1 (Listing The Properties Of A Field)

Suppose that $F$ is a field. Since $(F,+)$ and $(F^*,\cdot)$ are Abelian group, this means there are several properties that the binary operations $+$ and $*$ must satisfy. Make a list of all these properties, which together with the distributive laws, should give you a list of 9 properties that characterize a field. Then state at least two other sets that you know satisfy these properties.

Problem 2 (Algebraic Properties Of Modular Arithmetic)

Consider the sets $\mathbb{Z}_n$ for each positive integer $n$, together with modular addition and multiplication.

Give three different integers $n$ so that $\mathbb{Z}_n$ is a field.
Give an integer $n$ where $\mathbb{Z}_n$ is not a field. List the properties of being a field that are not satisfied.
Find an integer $n$ and elements $a,b\in \mathbb{Z}_n$ with $a\neq b$ such that $ab=0$ but neither $a$ nor $b$ is zero.
For each integer $k\geq 2$, find an integer $n$ and element $a\in \mathbb{Z}_n$ so that $a^k=0$ but $a^{k-1}\neq 0$.
State all $n$ for which $\mathbb{Z}_n$ is a field.

Definition (Polynomial Rings Over A Field)

Let $F$ be a field. We denote by $F[x]$ the set of all polynomial in the variable $x$ with coefficients in $F$ together with polynomial addition and multiplication (if needed, see page 294 in your text for a formal description of something you are very familiar with). The set $F[x]$ is called the polynomial ring over $F$ (in the indeterminate $x$). We can write each element $p(x)\in F[x]$ in the form $$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$$ for some $n$ where $a_i\in F$ for each $i$ and $a_n\neq 0$ if $n\geq 1$.

Problem 3 (Algebraic Properties Of Polynomial Rings Over A Field)

Let $F$ be a field (think the rationals $\mathbb{Q}$). Which of the properties of being a field does $F[x]$ not satisfy? Give an example of each property (of the 9 total) that is not satisfied.

The previous problem shows that polynomial rings over a field are not a field. Algebra's roots are in finding solutions to polynomial equation of the for $p(x)=0$. The number $\sqrt{2}$ is a solution to the equation $x^2-2=0$, and so is the root of some polynomial. We say that $\sqrt{2}$ is an algebraic number. The number $\pi$ is not the root of any polynomial (a very nontrivial thing to prove), and such numbers we call transcendental. They transcend algebra. One of our goals this semester will be to understand the set of algebraic numbers.

Definition (Algebraic Number)

We say that a number $x$ is algebraic if it is a root of a polynomial with rational coefficients. In symbols, we say that $x$ is algebraic if $x$ satisfies $p(x)=0$ for some $p(x)\in \mathbb{Q}[x]$.

Problem 4 (Algebraic Numbers)

Let $\mathbb{Z}[x]$ be the set of all polynomials with coefficients in $\mathbb{Z}$, together with the usual properties of polynomial addition and multiplication. Show that $x$ is an algebraic number if and only if $x$ satisfies $q(x)=0$ for some $q(x)\in \mathbb{Z}[x]$.

Let's now compare the field properties to operations with matrices and vectors. In the problems below, you'll be asked to list the properties of a field that are not satisfied. There are 9 properties to consider in each case, as there are 4 from each Abelian group and one more from the distributive laws. You should be prepared to prove each property that is satisfied (or at least state why you know it is satisfied).

Problem 5 (Algebraic Properties Of Square Matrices)

Let $\text{M}_2(\mathbb{Q})$ bet the set of 2 by 2 matrices with entries in $\mathbb{Q}$, together with the usual properties of matrix addition and matrix multiplication. Recall that $\text{GL}(2,\mathbb{Q})$ is the set of invertible 2 by 2 matrices with entries in $\mathbb{Q}$.

Which of the properties of being a field does $\text{M}_2(\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.
Which of the properties of being a field does $\text{GL}(2,\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.

Problem 6 (Algebraic Properties Of 3 D Vectors And The Cross Product)

Consider the set $\mathbb{R}^3$, together with the two binary operations of vector addition and the cross product. Which of the properties of being a field does $\mathbb{R}^3$ not satisfy? Give an example of each property that is not satisfied.

For fun, please see the wikipedia page on magmas. People have given lots of names to algebraic structures that satisfy various properties.

For more problems, see AllProblems

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