Extension Fields

Describe the elements of $\mathbb {Q}(\sqrt[3]{5})$.

Show that $\mathbb {Q}(\sqrt{2}, \sqrt{3}) = \mathbb {Q}(\sqrt{2}+\sqrt{3})$.

Find the splitting field of $x^{3}-1$ over $\mathbb {Q}$. Express your answer in the form $\mathbb {Q}(a)$.

Let $E$ be an extension field of $F$ and let $a$ and $b$ belong to $E$. Prove that $F(a, b) = F(a)(b) = F(b)(a)$.

Find the splitting field of $x^{4}+x^{2}+1 = (x^{2}+x+1)(x^{2}-x+1)$ over $\mathbb {Q}$.

Prove that $\mathbb {Q}(2^{1/3}) = \mathbb {Q}(2^{2/3})$.

Let $F$ be a field, and let $a$ and $b$ belong to $F$ with $a \neq 0$. If $c$ belongs to some extension field of $F$, prove that $F(c) = F(ac+b)$. ($F$ ``absorbs'' its own elements.)

Let $F = Z_{2}$ and let $f(x) = x^{3}+x+1 \in F[x]$. Suppose that $a$ is a zero of $f(x)$ in some extension field of $F$. How many elements does $F(a)$ have? Express each member of $F(a)$ in terms of $a$. Write out a complete multiplication table for $F(a)$.

Let $F(a)$ be the field described in Exercise 8. Express each of $a^{5}, a^{-2}$, and $a^{100}$ in the form $c_{2}a^{2}+c_{1}a+c_{0}$.

Let $F(a)$ be the field described in Exercise 8. Show that $a^{2}$ and $a^{2}+a$ are zeros of $x^{3}+x+1$.

Describe the elements in $\mathbb {Q}(\pi )$.

Let $F = \mathbb {Q}(\pi^{3})$. Find a basis for $F(\pi )$ over $F$.

Write $x^{7}-x$ as a product of linear factors over $Z_{3}$. Do the same for $x^{10}-x$.

Find all ring automorphisms of $\mathbb {Q}(\sqrt{5})$. Find all ring automorphisms of $\mathbb {Q}(\sqrt[3]{5})$.

Let $F$ be a field of characteristic $p$ and let $f(x) = x^{p}-a \in F[x]$. Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F$.

Suppose that $\beta$ is a zero of $f(x) = x^{4}+x+1$ in some extension field $E$ of $Z_{2}$. Write $f(x)$ as a product of linear factors in $E[x]$.

Find $a, b, c$ in $\mathbb {Q}$ such that $(1+\sqrt[3]{4})/(2-\sqrt[3]{2}) = a+b\sqrt[3]{2}+c\sqrt[3]{4}$.

Note that such $a, b, c$ exist, since $(1+\sqrt[3]{4})/(2-\sqrt[3]{2}) \in \mathbb {Q}(\sqrt[3]{2}) = \left\{ a+b\sqrt[3]{2}+c\sqrt[3]{4} \mid a, b, c \in \mathbb {Q}\right\}$.

Express $(3+4\sqrt{2})^{-1}$ in the form $a+b\sqrt{2}$, where $a, b \in \mathbb {Q}$.

Show that $\mathbb {Q}(4-i) = \mathbb {Q}(1+i)$, where $i = \sqrt{-1}$.

Find a polynomial $p(x)$ in $\mathbb {Q}^{x}$ such that $\mathbb {Q}(\sqrt{1+\sqrt{5}})$ is ring-isomorphic to $\mathbb {Q}^{x}/ \left\langle p(x)\right\rangle$.

Let $f(x) \in F[x]$ and let $a \in F$. Show that $f(x)$ and $f(x+a)$ have the same splitting field over $F$.

Recall that two polynomials $f(x)$ and $g(x)$ from $F[x]$ are said to be relatively prime if there is no polynomial of positive degree in $F[x]$ that divides both $f(x)$ and $g(x)$. Show that if $f(x)$ and $g(x)$ are relatively prime in $F[x]$, they are relatively prime in $K[x]$, where $K$ is any extension field of $F$.

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

Describe the elements of the extension field $\mathbb {Q}(\sqrt[4]{2})$ over the field $\mathbb {Q}(\sqrt{2})$.

What can you say about the order of the splitting field of $x^{5}+x^{4}+1 = (x^{2}+x+1)(x^{3}+x+1)$ over $Z_{2}$?

Find the splitting field of $x^{4}+1$ over $\mathbb {Q}$.

Write $x^{3}+2x+1$ as a product of linear polynomials over some extension field of $Z_{3}$.

Express $x^{8}-x$ as a product of irreducibles over $Z_{2}$.

Prove or disprove that $\mathbb {Q}(\sqrt{3})$ and $\mathbb {Q}(\sqrt{-3})$ are ring-isomorphic.

For any prime $p$, find a field of characteristic $p$ that is not perfect.

If $\beta$ is a zero of $x^{2}+x+2$ over $Z_{5}$, find the other zero.

Show that $x^{4}+x+1$ over $Z_{2}$ does not have any multiple zeros in any extension field of $Z_{2}$.

Show that $x^{21}+2x^{8}+1$ does not have multiple zeros in any extension of $Z_{3}$.

Theorem 19.5 (Criterion for Multiple Zeros): A polynomial $f(x)$ over a field $F$ has a multiple zero in some extension $E$ if and only if $f(x)$ and $f'(x)$ have a common factor of positive degree in $F[x]$.

Show that $x^{19}+x^{8}+1$ has multiple zeros in some extension of $Z_{3}$.

Let $F$ be a field of characteristic $p \neq 0$. Show that the polynomial $f(x) = x^{p^{n}}-x$ over $F$ has distinct zeros.

Theorem 19.5 (Criterion for Multiple Zeros): A polynomial $f(x)$ over a field $F$ has a multiple zero in some extension $E$ if and only if $f(x)$ and $f'(x)$ have a common factor of positive degree in $F[x]$.

Find the splitting field for $f(x) = (x^{2}+x+2)(x^{2}+2x+2)$ over $Z_{3}$. What is the order of the splitting field? Write $f(x)$ as a product of linear factors over $Z_{3}$.

Let $F$ be a field and $E$ an extension field of $F$ that contains $a_{1}, a_{2}, \ldots , a_{n}$. Prove that $F(a_{1}, a_{2}, \ldots , a_{n})$ is the intersection of all subfields of $E$ that contain $F$ and the set $\left\{ a_{1}, a_{2}, \ldots , a_{n}\right\}$. (This exercise is referred to in this chapter.)

Find the splitting field of $x^{4}-x^{2}-2$ over $Z_{3}$.

Suppose that $f(x)$ is a fifth-degree polynomial that is irreducible over $Z_{2}$. Prove that every nonidentity element is a generator of the cyclic group $(Z_{2}[x]/ \left\langle f(x)\right\rangle )^{*}$.

Show that $\mathbb {Q}(\sqrt{7}, i)$ is the splitting field for $x^{4}-6x^{2}-7$.

Suppose that $p(x)$ is a quadratic polynomial with rational coefficients and is irreducible over $\mathbb {Q}$. Show that $p(x)$ has two zeros in $\mathbb {Q}^{x}/ \left\langle p(x)\right\rangle$.

If $p(x) \in F[x]$ and $\deg p(x) = n$, show that the splitting field for $p(x)$ has at most $n!$ basis elements over $F$.

Let $p$ be a prime, $F = Z_{p}(t)$ (the field of quotients of the ring $Z_{p}[x]$) and $f(x) = x^{p}-t$. Prove that $f(x)$ is irreducible over $F$ and has a multiple zero in $K = F[x]/ \left\langle x^{p}-t\right\rangle$.

Let $f(x)$ be an irreducible polynomial over a field $F$. Prove that the number of distinct zeros of $f(x)$ in a splitting field divides $\deg f(x)$.

Let $n$ be an even positive integer and $F$ be a finite field of characteristic $2$. Explain why the factorization $x^{n}-x$ into irreducibles over $F$ cannot have the form

Let $F$ be a finite field of characteristic $2$ and $f(x)$ belong to $F[x]$. If the factorization of $f(x)$ into irreducibles over $F$ has the form $x(x-1)g_{1}(x)g_{2}(x) \cdots g_{k}(x)$ where the $g_{i}(x)$ are nonlinear, what is the maximum number of quadratic factors in the product?

What is the splitting field of $f(x) = x^{3}-2$ over $\mathbb {Q}(\sqrt[3]{2})$? What is the splitting field over $\mathbb {Q}(\sqrt{3}i)$?

Suppose that $a$ is a zero of $x^{3}-x-1$ in some extension field of $Z_{3}$. Write $(a^{2}+a)(a^{2}+1)$ in the form $a_{2}a^{2}+a_{1}a+a_{0}$ where $a_{2}, a_{1}, a_{0} \in Z_{3}$.

Does there exist an extension field $Z_{5}(\alpha )$ of $Z_{5}$ where $\alpha$ is not in $Z_{5}$ such that $Z_{5}(\alpha ) = \left\{ a\alpha +b \mid a, b \in Z_{5}\right\}$ and $\alpha^{2} = 2\alpha +1$? Give a reason for your answer.

Does there exist an extension field $F(\alpha )$ of $F$ where $\alpha$ is not in $F$ such that $F(\alpha ) = \left\{ a\alpha +b \mid a, b \in F\right\}$ with $\alpha^{2} = \alpha +2$? Give a reason for your answer.

Suppose that $x^{2}+1$ and $x^{2}+x+1$ have no zeros in a field $F$. If $\alpha$ is a zero of $x^{2}+1$ in some extension field $E$ of $F$ and $\beta$ is a zero of $x^{2}+x+1$ in $E$, prove that $\alpha$ is not in $F(\beta )$ and $\beta$ is not in $F(\alpha )$.

Give infinitely many examples of extension fields $F_{0}, F_{1}, F_{2}, \ldots$ of $\mathbb {Q}$ in which $x^{2}+1$ splits.