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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 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 $j$.

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 $F$.

  1. 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 $F$.)
  2. Show that the element $x+\left<p(x)\right>\in E$ is a zero of $p(x)$ in $E$.

Problem 65 (Every Element is a Product of Irreducibles in a PID)

Suppose that $D$ is a principle ideal domain with $a\in D$ where $a\neq 0$ is not a unit. Prove that $a$ can be written as the product of irreducibles.

Hint: You'll want to use the previous problem quite a bit. If $a$ is irreducible, then you're done. You'll probably want to start by showing that you can write $a=pc$ for some irreducible $p$ (in other words, you can factor off an irreducible. How will you use the previous problem? If you can't factor off an irreducible, then you can write $a=a_1b_1$ for two non unit non irreducible terms. Why do we know $\langle a\rangle\subseteq\langle a_1\rangle$? Now repeat this process to get $\langle a\rangle\subseteq\langle a_1\rangle\subseteq\langle a_2\rangle$ and so on. Once you've gotten $a=pc$ for some irreducible $p$, what connection is there between $\langle a\rangle$ and $\langle c\rangle$?

Problem 66 (Every Polynomial Has A Root In Some Extension Field)

Let $F$ be a field and $f(x)\in F[x]$ be nonzero and not a unit. Prove that there exists an extension field $E$ in which $f(x)$ has a zero.

Problem 68 (Multiple Zeros and The Derivative Proof)

Suppose that $F$ is field and $f(x)\in F[x]$. Prove that $f(x)$ has a zero of multiplicity greater than one in some extension field $E$ if and only if $f$ and $f'$ have a common factor in $F[x]$.

Problem 69 (Showing The Existence of a Field Extension Generated by a subset)

Suppose that $E$ is an extension field of the field $F$. Let $S$ be a subset of $E$.

  1. Prove that there is a subfield of $E$ that contains both $F$ and $S$.
  2. Prove that the intersection of all subfields of $E$ that contain both $F$ and $S$ is a subfield of $E$.

Problem 70 (Factoring By An Irreducible Gives A Simple Extension)

Let $F$ be a field and $p(x)\in F[x]$ be irreducible over $F$. If $a$ is a zero of $p(x)$ in some extension $E$ of $F$, then $F(a)$ is isomorphic to $F[x]/\left<p(x)\right>$. Furthermore, if $\deg p(x) = n$, then every member of $F(a)$ can be uniquely expressed in the form $c_{n-1}a^{n-1} + c_{n-2}a^{n-2} + \cdots + c_{1}a + c_{0}$ where $c_i\in F$. (In other words, the set $\left\{1, a, a^2, \ldots, a^{n-1}\right\}$ is a basis for $F(a)$ over $F$).

Problem 71 (Simple Extensions For The Same Polynomial Are Isomorphic)

Let $F$ be a field and let $p(x)\in F[x]$ be irreducible over $F$. If $a$ is a zero of $p(x)$ in some extension $E$ of $F$ and $b$ is a zero of $p(x)$ in some extension $E'$ of $F$, then the fields $F(a)$ and $F(b)$ are isomorphic.

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.

Problem 73 (Unique Factorization In A PID)

Suppose that $D$ is a principal ideal domain. Suppose that $f=p_1p_2\cdots p_m = q_1q_2\cdots q_n,$ where each $p_i$ and $q_i$ is irreducible over $D$. Prove that $n=m$ and after rearranging, for each $i$ we have $p_i=u_iq_i$ for some unit $i$. In other words, prove that $D$ uniquely factors as a product of irreducibles.

For more problems, see AllProblems