Abstract Algebra: Theory and Applications

Compute each of the following.

Use the division algorithm to find $q(x)$ and $r(x)$ such that $a(x)=q(x)b(x)+r(x)$ with $\mathrm{deg}r(x)<\mathrm{deg}b(x)$ for each of the following pairs of polynomials.

Find the greatest common divisor of each of the following pairs $p(x)$ and $q(x)$ of polynomials. If $d(x)=gcd(p(x),q(x))\text{,}$ find two polynomials $a(x)$ and $b(x)$ such that $a(x)p(x)+b(x)q(x)=d(x)\text{.}$

Find all of the zeros for each of the following polynomials.

Which of the following polynomials are irreducible over $\mathbb{Q}[x]\text{?}$

Let

where $a_n\ne 0\text{.}$ Prove that if $p(r/s)=0\text{,}$ where $gcd(r,s)=1\text{,}$ then $r\mid a_0$ and $s\mid a_n\text{.}$

The polynomial

for $p$ prime is called the cyclotomic polynomial. Show that ${\mathrm{\Phi }}_p(x)$ is irreducible over $\mathbb{Q}$ for any prime $p\text{.}$

Show that $x^p-x$ has $p$ distinct zeros in ${\mathbb{Z}}_p\text{,}$ for any prime $p\text{.}$ Conclude that

Let $F$ be a field and $f(x)=a_0+a_{1}x+\cdots +a_n{x}^n$ be in $F[x]\text{.}$ Define $f^{\prime }(x)=a_1+2a_{2}x+\cdots +n{a}_n{x}^{n-1}$ to be the derivative of $f(x)\text{.}$