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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 $E$.
For more problems, see AllProblems