Math 441 - Abstract Algebra - Schedule

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February 3

Please work on the division algorithm. We'll be presenting it on Monday. Also, complete your 3rd write up before class. You can pick any problem from week 4, or one from week 3.

We'll finish up any proofs we didn't finish from Friday (So Sam's got #5 and Nick/Others have #2). Then we'll look at some examples of $Z[\sqrt{d}]$ and $Q[\sqrt{d}]$, as well as $Z_p[\sqrt{d}]$. We'll also spend some more time looking at factor rings over IDs versus fields. I would like you to see examples of integral domains where unique factorization breaks down (so solving algebraic equations becomes nightmarish).

Construct a field with 8 elements.

February 5

Start by completing the division algorithm for $\mathbb{Q}[x]$. Then work on the following.

Problem (When does $(x+y)^n=x^n+y^n$)

Let $R$ be an integral domain with characteristic $n$. Compute $(x+y)^n$ for $n=1,2,3,4,5,6,7$. For which $n$ does $(x+y)^n=x^n+y^n$? Make a conjecture.

Problem 44 (Not Every Subring Is An Ideal)

Give an example of a ring $R$ and a subring $S$ so that $S$ is not an ideal of $R$. Make sure you prove that $S$ is a subring, but not an ideal.

Problem 45 (A Homomorphism From Z To A Ring With Unity)

Let $R$ be a ring with unity 1. After you have finished the first step below, you should be able to give answers to the remaining parts by referring to the first part and using the properties of ring homomorphisms.

Show that the mapping $\phi:\mathbb{Z}\to R$ given by $\phi(n)=n\cdot 1$ is a ring homomorphism.
Show that if $R$ has characteristic $n>0$, then $R$ contains a subring isomorphic to $Z_n$.
Show that if $R$ has characteristic $n=0$, then $R$ contains a subring that is isomorphic to $Z$.
Show that for any positive integer $m$, the mapping of $\phi:\mathbb{Z}\to \mathbb{Z}_m$ given by $x\to x$ mod $m$ is a ring homomorphism.

Problem 46 (Every Field Contains A Subfield Isomorphic To $\mathbb{Z}_p$ or $\mathbb{Q}$)

Suppose that $F$ is a field.

If $F$ has prime characteristic $p$, show that $F$ contains a subfield isomorphic to $Z_p$.
If $F$ has characteristic 0, show that $F$ contains a subfield isomorphic to $\mathbb{Q}$.

Hint:You'll want to use the previous problem. If $p$ is prime, the previous problem should give this quickly. If the characteristic is zero, you'll need to use the copy of $Z$ sitting inside $F$ to create the subfield $Q$.

Problem 47 (The Remainder Theorem)

Let $F$ be a field, $a\in F$, and $f(x)\in F[x]$. Prove that $f(a)$ is the remainder in division of $f(x)$ by $x-a$.

Problem 48 (The Factor Theorem)

Let $F$ be a field, $a\in F$, and $f(x)\in F[x]$. Prove that $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.

February 7

Problem 49 (Polynomials of degree $n$ have at most $n$ zeros)

Show that a polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity.

Problem 50 (We have $I=\left<g(x)\right>$ if and only if $g(x)$ is a polynomial of minimal degree in $I$)

Let $F$ be field and $I$ a nonzero ideal in $F[x]$, and $g(x)$ an element of $F[x]$. Show that $I=\langle g(x)\rangle$ if and only if $g(x)$ is a nonzero polynomial of minimal degree in $I$.

In the previous problem, we showed that any ideal in $F[x]$ is generated by a single polynomial. That's pretty remarkable, in that no matter what polynomials we use to generate an ideal, we can always find a single polynomial that generates the whole ideal. This property turns out to be extremely useful, and as such we'll give any ring with this property a special name.

Definition (Principal Ideal Domain PID)

A principal ideal domain (PID) is an integral domain in which every ideal has the form $\left<a\right> = \{ra|r\in R\}$ for some $a$ in $R$.

We know that $F[x]$ is a principle ideal domain provided $F$ is a field. Are there other principle ideal domains?

Problem 51 (Polynomial Rings Over PIDs need not be PIDs)

Show that $\mathbb{Z}$ is a principle ideal domain. Then show that $\mathbb{Z}[x]$ is not a principle ideal domain.

Problem 52 (The Degree Of A Product Of Polynomials)

Suppose that $D$ is an integral domain, and suppose that $f(x),g(x)\in D[x]$.

Prove that $\deg(f(x)\cdot g(x)) = \deg(f(x))+\deg(g(x))$.
Then give an example of a commutative ring $R$ and two polynomials so that $\deg(f(x)\cdot g(x)) < \deg(f(x))+\deg(g(x))$.

February 10

We'll show that $R[x]/<x^2+1>\approx C$.

We'll also be finishing up several problems from other days.

February 12

We'll be finishing up Darrel's proof that a polynomial of degree $n$ can have at most $n$ zeros, counting multiplicity. We'll also have Tara share her example of a PID such that the polynomial ring is not a PID. Brennan's excited to share his work on last Wednesday's #4. Then we'll work on as many of the following as there is time.

We'd like to know when we obtain polynomials $p(x)\in F[x]$ such that $F[x]/\left<p(x)\right>$ is a field. We already know this means that $I=\left<p(x)\right>$ must be a maximal ideal in $F[x]$. We've also seen that if we can factor $p(x)$ as the product of two polynomials (that don't have multiplicative inverses), then $I$ is not maximal. Let's make a definition to make this precise.

Definition (Irreducible Polynomial Reducible Polynomial)

Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product $f(x)=g(x)h(x)$, with $g(x)$ and $h(x)$ from $D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$. A nonzero, nonunit element of $D[x]$ that is not irreducible over $D$ is called reducible over $D$.

In general, it's a hard problem to determine when a polynomial is reducible or irreducible. However, there are a few cases where it's easy.

Problem 53 (Reducibility Test For Degrees 2 And 3)

Let $F$ be a field. If $f(x)\in F[x]$ and deg $f$ is 2 or 3, then prove that $f(x)$ is reducible over $F$ if and only if $f(x)$ has a zero in $F$.

Before we look at any other reducibility tests, let's prove a key theorem, namely that $p(x)$ is irreducible if and only if $\left<p(x)\right>$ is maximal. We really want to know when we can guarantee that $F[x]/\left<p(x)\right>$ is a field.

Problem 54 (We Know $\left<p(x) \right>$ Is Maximal Iff $p(x)$ Is Irreducible)

Let $F$ be a field and let $p(x)\in F[x]$. Prove that $\left<p(x) \right>$ is a maximal ideal if and only if $p(x)$ is irreducible over $F$.

Now let's look at some other tests for reducibility. This next definition just gets rid of a complication from some polynomials, by removing from the polynomial any common factors.

Definition (Content Of A Polynomial Primitive Polynomial)

The content of a polynomial in $\mathbb{Z}[x]$ is the greatest common divisor of the coefficients. A primitive polynomial has content 1.

Problem 54.5 (The Product Of Primitives Is Primitive)

Prove that the product of two primitive polynomials is primitive.

As you work through the problems in the next few weeks, you'll want to pay close attention to the assumptions. In some problems we assume that we are working in a field. In some problems, we assume that we are working in an integral domain. The next problem shows that if you can show something is reducible over the field $\mathbb{Q}$, then it is reducible over $\mathbb{Z}$.

Problem 55 (Reducibility Over Q Implies Reducibility Over Z)

Let $f(x)\in \mathbb{Z}[x]$. Prove that if $f(x)$ is reducible over $\mathbb{Q}$, then $f(x)$ is reducible over $\mathbb{Z}$.

The contrapositive to the previous problem is extremely powerful, namely if a polynomial with integer coefficients is not reducible over $\mathbb{Z}$, then it is not reducible over $\mathbb{Q}$. For this reason, we'll now study irreducibility tests over $\mathbb{Z}$.

Problem 56 (Mod P Irreducibility Test)

Let $p$ be a prime and suppose that $f(x)\in \mathbb{Z}[x]$. Let $\bar f (x)$ be the polynomial in $\mathbb{Z}_p[x]$ obtained by reducing the coefficients of $f(x)$ modulo $p$. Prove that if if $\bar f (x)$ is irreducible over $\mathbb{Z}_p$ and $\text{deg }\bar f(x) = \text{deg }f(x)$, then $f(x)$ is irreducible over $\mathbb{Q}$.

Problem 57 (Eisenstein's Criterion)

Let $f(x)=a_nx^n + \cdots +a_1x +a_0$. Prove that if there is a prime $p$ such that $p$ divides every coefficient but $a_n$ and $p^2$ does not divide $a_0$, then $f(x)$ is irreducible over $\mathbb{Q}$.

February 14

Definition (Irreducible Polynomial Reducible Polynomial)

In general, it's a hard problem to determine when a polynomial is reducible or irreducible. However, there are a few cases where it's easy.

Problem 53 (Reducibility Test For Degrees 2 And 3)

Let $F$ be a field. If $f(x)\in F[x]$ and deg $f$ is 2 or 3, then prove that $f(x)$ is reducible over $F$ if and only if $f(x)$ has a zero in $F$.

Problem 54 (We Know $\left<p(x) \right>$ Is Maximal Iff $p(x)$ Is Irreducible)

Let $F$ be a field and let $p(x)\in F[x]$. Prove that $\left<p(x) \right>$ is a maximal ideal if and only if $p(x)$ is irreducible over $F$.

Problem 55 (Reducibility Over Q Implies Reducibility Over Z)

Let $f(x)\in \mathbb{Z}[x]$. Prove that if $f(x)$ is reducible over $\mathbb{Q}$, then $f(x)$ is reducible over $\mathbb{Z}$.

Problem 56 (Mod P Irreducibility Test)

Problem 57 (Eisenstein's Criterion)

Problem 58 (Rational Root Test)

Suppose that $$f(x) = a_nx^n+\cdots +a_1x+a_0\in \mathbb{Z}[x],$$ with $a_n\neq 0$. Prove that if $r$ and $s$ are relatively prime and $f(r/s)=0$, then we must have $r\mid a_0$ and $s\mid a_n$.

Problem 59 (Irreducibles Behave Like Prime Numbers)

Let $F$ be a field and suppose that $p(x)\in F[x]$ is irreducible over $F$. Suppose also that $p(x)$ divides the product $a_1(x)a_2(x)\cdots a_n(x)$ where $a_i(x)\in F[x]$ for each $i$. Prove that $p(x)$ must divide $a_k(x)$ for some $k$.

February 17

February 19

We'll be focusing on irreducibility of polynomials today. I'll also have you prove several corrollaries, small lemmas, etc., as groups in class.

Enjoy your break. This is week 7, but we'll combine week 6 and 7 online posts. So by the end of this week, you should have 5 posts up.

Br. Woodruff

Problem 55 (Reducibility Over Q Implies Reducibility Over Z)

Let $f(x)\in \mathbb{Z}[x]$. Prove that if $f(x)$ is reducible over $\mathbb{Q}$, then $f(x)$ is reducible over $\mathbb{Z}$.

Problem 56 (Mod P Irreducibility Test)

Problem 57 (Eisenstein's Criterion)

Problem 58 (Rational Root Test)

Suppose that $$f(x) = a_nx^n+\cdots +a_1x+a_0\in \mathbb{Z}[x],$$ with $a_n\neq 0$. Prove that if $r$ and $s$ are relatively prime and $f(r/s)=0$, then we must have $r\mid a_0$ and $s\mid a_n$.

Problem 59 (Irreducibles Behave Like Prime Numbers)

February 21

If we know $f(x) = (x-a)^k\cdot q(x)$, and $b\neq a$ is a zero of $q$, prove that $b$ is a zero of $f$ with the exact same multiplicity. Why do we need to prove this? What does this tell us about unique factorization.

Let's practice testing if polynomials are reducible or irreducible over $\mathbb{Q}$. These polynomials come straight out of Gallian.

$21x^3-3x^2+2x+9$
$x^5+2x+4$ (check mod 2 and mod 3 - we'll have 9 cases to check. Divide up the work.)
$3x^5+15x^4-20x^3+10x+20$
$x^5+9x^4+12x^2+6$
$x^4+x+1$
$x^4+3x^2+3$
$(5/2)x^5 +(9/2)x^4+15 x^3+(3/7(x^2+6x + 3/14$

Show that $x^4+1$ is irreducible over $\mathbb{Q}$, but reducible over $\mathbb{Z}_p$ for every prime $p$.

Find all the zeros of $3x^2+x+4$ over $\mathbb{Z}_7$. Do so in two ways. First, find all the zeros by substitution. Then use the quadratic formula. Then repeat this with $2x^2+x+3$ over $\mathbb{Z}_5$. When does the quadratic formula work?

Show that the ideal $\left<x^2+1\right>$ is prime in $\mathbb{Z}[x]$ but not maximal in $\mathbb{Z}[x]$.

February 24

February 26

Problem (Additional Properties Of Irreducible Polynomials Over A Field)

Let $F$ be a field and $p(x)$ an irreducible polynomial over $F$.

Prove that $F[x]/\left<p(x)\right>$ is a field.
Let $a(x),b(x)\in F[x]$. If $p(x)$ divides $a(x)b(x)$, then prove that $p(x)$ divides $a(x)$ or $p(x)$ divides $b(x)$.

Theorem (Unique Factorization In ZX)

Every polynomial that is not the zero polynomial or a unit in $Z[x]$ can be written in the form $b_1b_2\cdots b_sp_1(x)p_2(x)\cdots p_m(x)$, where the $b_i$'s are irreducible polynomials of degree 0, and the $p_i(x)$'s are irreducible polynomials of positive degree. Furthermore, if we completely factor in 2 ways, $$b_1b_2\cdots b_sp_1(x)p_2(x)\cdots p_m(x)=c_1c_2\cdots c_tq_1(x)q_2(x)\cdots q_n(x),$$ then $s=t$, $m=n$, and after renumbering the $c$'s and $q(x)$'s, we have $b_i=\pm c_i$ and $p_j(x)=\pm q_j(x)$ for all $i$ and $j$.

Problem 60 (Unique Factorization Existence Proof)

Prove that every polynomial that is not the zero polynomial or a unit in $Z[x]$ can be written in the form $b_1b_2\cdots b_sp_1(x)p_2(x)\cdots p_m(x)$, where the $b_i$'s are irreducible polynomials of degree 0, and the $p_i(x)$'s are irreducible polynomials of positive degree.

Problem 61 (Unique Factorization Uniqueness Proof)

Suppose that we have factored a polynomial in $\mathbb{Z}[x]$ in two way, namely $$b_1b_2\cdots b_sp_1(x)p_2(x)\cdots p_m(x)=c_1c_2\cdots c_tq_1(x)q_2(x)\cdots q_n(x).$$ Prove that $s=t$, $m=n$, and after renumbering the $c$'s and $q(x)$'s, we have $b_i=\pm c_i$ and $p_j(x)=\pm q_j(x)$ for all $i$ and $j$.

Definition (Associates Irreducibles Primes)

Let $D$ be an integral domain. All elements below are elements of $D$.

We say that two elements $a$ and $b$ are associates if $a=ub$ for some unit $u$.
If $a$ is nonzero and not a unit, then we say $a$ is an irreducible if whenever $a=bc$, then either $b$ or $c$ is a unit.
If $a$ is nonzero and not a unit, then we say $a$ is a prime if $a\mid bc$ implies $a\mid b$ or $a\mid c$.

Problem 62 (Prime Implies Irreducible)

In an integral domain, prove that if an element $a$ is prime, then $a$ must be irreducible.

Problem 63(Irreducible Polynomials Have A Zero In Some Extension Field)

Let $F$ be a field and suppose that $p(x)$ is an irreducible polynomial over $F$.

Prove that $E=F[x]/\left<p(x)\right>$ is an extension field of $F$. (Note: An extension field of $F$ is a field $E$ such that $E$ contains a subfield isomorphic to $F$.)
Show that the element $x+\left<p(x)\right>\in E$ is a zero of $p(x)$ in $E$.

February 28

IrreduciblesPolynomialsBehaveLikePrimeNumbersOverTheIntegers

We know that if we were in $\mathbb{Q}$ instead of $\mathbb{Z}$, then the problems we have worked on before make this trivial. So what connection is there between reducibility over $\mathbb{Z}$ and $\mathbb{Q}$. We've work on problems that give exactly what we need.

Definition (Associates Irreducibles Primes)

Let $D$ be an integral domain. All elements below are elements of $D$.

We say that two elements $a$ and $b$ are associates if $a=ub$ for some unit $u$.
If $a$ is nonzero and not a unit, then we say $a$ is an irreducible if whenever $a=bc$, then either $b$ or $c$ is a unit.
If $a$ is nonzero and not a unit, then we say $a$ is a prime if $a\mid bc$ implies $a\mid b$ or $a\mid c$.

Problem 62 (Prime Implies Irreducible)

In an integral domain, prove that if an element $a$ is prime, then $a$ must be irreducible.

Problem 72 (Irreducible Implies Prime in a PID)

We've already shown that every prime is irreducible. The converse is not always true. However, suppose that $D$ is a principle ideal domain. Prove that if an element $a$ is irreducible then it is prime.

Definition (Associates Irreducibles Primes)

Let $D$ be an integral domain. All elements below are elements of $D$.

We say that two elements $a$ and $b$ are associates if $a=ub$ for some unit $u$.
If $a$ is nonzero and not a unit, then we say $a$ is an irreducible if whenever $a=bc$, then either $b$ or $c$ is a unit.
If $a$ is nonzero and not a unit, then we say $a$ is a prime if $a\mid bc$ implies $a\mid b$ or $a\mid c$.

Problem 63(Irreducible Polynomials Have A Zero In Some Extension Field)

Let $F$ be a field and suppose that $p(x)$ is an irreducible polynomial over $F$.

Prove that $E=F[x]/\left<p(x)\right>$ is an extension field of $F$. (Note: An extension field of $F$ is a field $E$ such that $E$ contains a subfield isomorphic to $F$.)
Show that the element $x+\left<p(x)\right>\in E$ is a zero of $p(x)$ in $E$.

For more problems, see AllProblems