Extension Fields and Splitting Fields

Verify that the field $F$ in 1.1 is indeed isomorphic to $\mathbb{C}$ by using the map $\phi$ given. For completeness, also verify that $\phi$ is well-defined.

Let $f(x)=2x+1\in\mathbb{Z}_{4}[x]$. Prove that no ring that contains a subring that is isomorphic to $\mathbb{Z}_{4}$ has a zero of $f(x)$.

Find two extension fields that contains a zero of the polynomial $x^{4}+3x^{2}+1\in\mathbb{Q}[x]$.

Prove 1.22.

Let $f(x)=x^{4}-5x^{2}+6\in\mathbb{Q}[x]$.

1. Show that one possible splitting field of $f(x)$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})(\sqrt{3})$.

2. It is given that $p(x)=x^{4}-10x+1$ is an irreducible polynomial over $\mathbb{Q}$. Show that $p(x)$ has a zero of $\sqrt{2}+\sqrt{3}$.

3. Hence show that $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ is a splitting field of $f(x)$ over $\mathbb{Q}$.

Prove for any field $F$ and any $f(x),g(x)\in F[x]$ that $(f(x)+g(x))^{\prime}=f^{\prime}(x)+g^{\prime}(x)$.

Prove for any field $F$ and any polynomial $f(x)\in F[x]$ that

where $n$ is a positive integer.

Let $F$ be a field and let $f(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}\in F[x]$. Prove that $f^{\prime}(x)=0$ implies $ra_{r}=0$ for all $r\in\{1,2,\dots,n\}$.

Let $F$ be a field, and let $p,q\in F$ where $p\neq 0$. Suppose $E/F$ is a field extension and $a\in E$. Prove that $F(pa+q)=F(a)$.

Determine which of the following polynomials have multiple zeroes over the field $F$ specified. If the polynomial has a multiple zero, state its value.

1. $x^{4}+2x+7$ over $F=\mathbb{Z}_{2}$.

2. $x^{19}+x^{8}+1$ over $F=\mathbb{Z}_{3}$.

3. $2x^{6}+x^{4}+2x^{3}+2$ over $F=\mathbb{Z}_{3}$.

4. $x^{8}+3x^{5}+x^{3}+5$ over $F=\mathbb{Z}_{7}$.

Let $p(x)\in\mathbb{Z}_{2}$ be a polynomial of degree 7 that is irreducible in $\mathbb{Z}_{2}$. Prove or disprove the following statements.

1. $\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle$ is a field.

2. $\left(\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle\right)^{\ast}$ is a group under coset multiplication.

3. Every non-identity element in $\left(\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle\right)^{\ast}$ is a generator.

Find all subfields of $\mathbb{Q}(\sqrt{2})$.

Let $F$ be a field of prime characteristic $p$. Prove that $f(x)=x^{p^{n}}-x\in F[x]$, where $n$ is a positive integer, only has simple zeroes.

Let $F$ be a field of prime characteristic $p$, and let $a\in F$. Prove that $f(x)=x^{p}-a\in F[x]$ is either irreducible or splits over $F$.(Hint: consider two cases – either $f(x)$ has a zero in $F$ or it does not.)