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Today

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 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 $k$.

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 $F$.

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 $E$.

For more problems, see AllProblems