Chapter Questions
Prove Theorem $21.2$ and Theorem $21.3 .$
Let $E$ be the algebraic closure of $F$. Show that every polynomial in $F[x]$ splits in $E$.
Prove that $Q(\sqrt{2}, \sqrt[3]{2}, \sqrt[4]{2}, \ldots)$ is an algebraic extension of $Q$ but not a finite extension of $Q .$ (This exercise is referred to in this chapter.)
Let $E$ be an algebraic extension of $F$. If every polynomial in $\overline{F[x]}$ splits in $E$, show that $E$ is algebraically closed.
Suppose that $F$ is a field and every irreducible polynomial in $\bar{F}[x]$ is linear. Show that $F$ is algebraically closed.
Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that deg $\overline{f(x)}$ and deg $g(x)$ are relatively prime. If $a$ is a zero of $f(x)$ in some $e x-$ tension of $F$, show that $g(x)$ is irreducible over $F(a)$.
Let $a$ and $b$ belong to $Q$ with $b \neq 0$. Show that $Q(\sqrt{a})=Q(\sqrt{b})$ if and only if there exists some $c \in Q$ such that $a=b c^{2}$.
Find the degree and a basis for $Q(\sqrt{3}+\sqrt{5})$ over $Q(\sqrt{15})$. Find the degree and a basis for $Q(\sqrt{2}, \sqrt[3]{2}, \sqrt[4]{2})$ over $Q$.
Suppose that $E$ is an extension of $F$ of prime degree. Show that, for every $a$ in $E, F(a)=F$ or $F(a)=E$.
If $[F(a): F]=5$, find $\left[F\left(a^{3}\right): F\right] .$ Does your argument apply equally well if $a^{3}$ is replaced with $a^{2}$ or $a^{4}$ ?
Without using the Primitive Element Theorem, prove that if $[K: F]$ is prime, then $K$ has a primitive element.
Let $a$ be a complex number that is algebraic over $Q .$ Show that $\sqrt{a}$ is algebraic over $Q$.
Let $\beta$ be a zero of $f(x)=x^{5}+2 x+4$ (see Example 8 in Chapter 17). Show that none of $\sqrt{2}, \sqrt[3]{2}, \sqrt[4]{2}$ belongs to $Q(\beta)$.
Prove that $Q(\sqrt{2}, \sqrt[3]{2})=Q(\sqrt[6]{2})$.
Let $a$ and $b$ be rational numbers. Show that $Q(\sqrt{a}, \sqrt{b})=$ $Q(\sqrt{a}+\sqrt{b})$
Find the minimal polynomial for $\sqrt[3]{2}+\sqrt[3]{4}$ over $Q$.
Let $K$ be an extension of $F$. Suppose that $E_{1}$ and $E_{2}$ are contained in $K$ and are extensions of $F$. If $\left[E_{1}: F\right]$ and $\left[E_{2}: F\right]$ are both prime, show that $E_{1}=E_{2}$ or $E_{1} \cap E_{2}=F$
Let $a$ be a nonzero algebraic element over $F$ of degree $n .$ Show that $a^{-1}$ is also algebraic over $F$ of degree $n .$
Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$.
If $a b$ is algebraic over $F$ and $b \neq 0$, prove that $a$ is algebraic over $F(b)$.
Let $E$ be an algebraic extension of a field $F$. If $R$ is a ring and $E \supseteq$ $R \supseteq F$, show that $R$ must be a field.
Prove that $\pi^{2}-1$ is algebraic over $Q\left(\pi^{3}\right)$.
If $a$ is transcendental over $F$, show that every element of $F(a)$ that is not in $F$ is transcendental over $F$.
Suppose that $E$ is an extension of $F$ and $a, b \in E$. If $a$ is algebraic over $F$ of degree $m$, and $b$ is algebraic over $F$ of degree $n$, where $m$ and $n$ are relatively prime, show that $[F(a, b): F]=m n .$
Let $K$ be a field extension of $F$ and let $a \in K .$ Show that $\left[F(a): F\left(a^{3}\right)\right] \leq 3 .$ Find examples to illustrate that $\left[F(a): F\left(a^{3}\right)\right]$ can be 1,2, or 3 .
Find an example of a field $F$ and elements $a$ and $b$ from some extension field such that $F(a, b) \neq F(a), F(a, b) \neq F(b)$, and $[F(a, b): F]$ $<[F(a): F][F(b): F]$
Let $E$ be a finite extension of $\mathbf{R}$. Use the fact that $\mathbf{C}$ is algebraically closed to prove that $E=\mathbf{C}$ or $E=\mathbf{R}$.
Suppose that $[E: Q]=2 .$ Show that there is an integer $d$ such that $E=Q(\sqrt{d})$ where $d$ is not divisible by the square of any prime.
Suppose that $p(x) \in F[x]$ and $E$ is a finite extension of $F$. If $p(x)$ is irreducible over $F$, and deg $p(x)$ and $[E: F]$ are relatively prime, show that $p(x)$ is irreducible over $E$.
Let $E$ be an extension field of $F$. Show that $[E: F]$ is finite if and only if $E=F\left(a_{1}, a_{2}, \ldots, a_{n}\right)$, where $a_{1}, a_{2}, \ldots, a_{n}$ are algebraic over $F$.
If $\alpha$ and $\beta$ are real numbers and $\alpha$ and $\beta$ are transcendental over $Q$, show that either $\alpha \beta$ or $\alpha+\beta$ is also transcendental over $Q$.
Let $f(x)$ be a nonconstant element of $F[x] .$ If $a$ belongs to some extension of $F$ and $f(a)$ is algebraic over $F$, prove that $a$ is algebraic over $F$.
Let $f(x)=a x^{2}+b x+c \in Q[x] .$ Find a primitive element for the splitting field for $f(x)$ over $Q$.
Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a$ and $b$ belong to some extension $E$ of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$.
Let $f(x) \in F[x]$. If deg $f(x)=2$ and $a$ is a zero of $f(x)$ in some extension of $F$, prove that $F(a)$ is the splitting field for $f(x)$ over $F$.
Let $a$ be a complex zero of $x^{2}+x+1$ over $Q .$ Prove that $Q(\sqrt{a})=Q(a)$
If $F$ is a field and the multiplicative group of nonzero elements of $F$ is cyclic, prove that $F$ is finite.
Let $a$ be a complex number that is algebraic over $Q$ and let $r$ be a rational number. Show that $a^{r}$ is algebraic over $Q$.
Prove that, if $K$ is an extension field of $F$, then $[K: F]=n$ if and only if $K$ is isomorphic to $F^{\text {i }}$ as vector spaces. (See Exercise 27 in Chapter 19 for the appropriate definition. This exercise is referred to in this chapter.)
Let $a$ be a positive real number and let $n$ be an integer greater than 1 . Prove or disprove that $\left[Q\left(a^{1 / n}\right): Q\right]=n$.
Let $a$ and $b$ belong to some extension field of $F$ and let $b$ be algebraic over $F$. Prove that $[F(a, b): F(a)] \leq[F(a, b): F]$.
Let $F, K$, and $L$ be fields with $F \subseteq K \subseteq L$. If $L$ is a finite extension of $F$ and $[L: F]=[L: K]$, prove that $F=K$.
Let $F$ be a field and $K$ a splitting field for some nonconstant polynomial over $F$. Show that $K$ is a finite extension of $F$.
Prove that $\mathbf{C}$ is not the splitting field of any polynomial in $Q[x]$.
Prove that $\sqrt{2}$ is not an element of $Q(\pi)$.
Let $\alpha=\cos \frac{2 \pi}{7}+i \sin \frac{2 \pi}{7}$ and $\beta=\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5} .$ Prove that $\beta$is not in $Q(\alpha)$.
Let $m$ be a positive integer. If $a$ is transcendental over a field $F$, prove that $a^{m}$ is transcendental over $F$.
Suppose $K$ is an extension of $F$ of degree $n$. Prove that $K$ can be written in the form $F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ for some $x_{1}, x_{2}, \ldots, x_{0}$ in $K .$
Prove that there are no positive integers $m$ and $n$ such that $\sqrt{2}^{m}=\pi^{n}$.