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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 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 $j$.

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 64 (Ascending Chains of Ideals In a PID Are Finite)

Suppose that $R$ is a ring, and that $A_1\subseteq A_2\subseteq A_3\subseteq \cdots$ is nested sequence of ideals of $R$.

  1. Prove that the union $\ds \bigcup_{n=1}^{\infty} A_n$ is an ideal of $R$.
  2. If $R$ is a principle ideal domain, prove that there exists an integer $k$ so that $A_k=A_n$ for every $n\geq k$. In other words, prove that every ascending chain of ideals in a principle ideal domain contain finitely many different ideals.
  3. Challenge: Give an example of a ring $R$ and an infinite ascending chain of ideals so that no two ideals are equal.

You can present in class even if you don't get the third part.

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 67 (The Sum and Product Rule for Derivatives)

Suppose that $F$ is a field and that $f(x),g(x)\in F[x]$.

  1. Prove the sum rule, namely that $(f+g)'(x)=f'(x)+g'(x)$.
  2. Prove the product rule, namely that $(fg)'(x)=f'(x)g(x)+f(x)g'(x)$.
  3. What properties of the field $F$ were needed to complete your proofs? State a stronger theorem that requires less assumptions.

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.

For more problems, see AllProblems