Please Login to access more options.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
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 $F$.
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 $\mathbb{Q}$.
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 $k$.
For more problems, see AllProblems