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This next definition just gets rid of a complication from some polynomials, by removing from the polynomial any common factors.
Definition (Content Of A Polynomial Primitive Polynomial)
The content of a polynomial in $\mathbb{Z}[x]$ is the greatest common divisor of the coefficients. A primitive polynomial has content 1.
Problem 54.5 (The Product Of Primitives Is Primitive)
Prove that the product of two primitive polynomials is primitive.
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 $\mathbb{Z}$.
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