Fields and Their Extensions

Let $R$ be the set of $2 \times 2$ matrices of the form

where $a, b$ are elements of a given field $F$. Show that with the usual matrix operations, $R$ is a commutative ring with identity. For which of the following fields $F$ is $R$ a field: $F = \mathbb {Q}, \mathbb {C}, \mathbb {Z}^{5}, \mathbb {Z}^{7}$?

Prove that every finite integral domain is a field.

Let $F \subset K$ be fields, and $a$ and $b$ elements of $K$ which are algebraic over $F$. Show that $a+b$ is algebraic over $F$.

Prove that every finite multiplicative group of complex numbers is cyclic.

Let $F$ be a field. Prove that every finite subgroup of the multiplicative group of nonzero elements of $F$ is cyclic.

Let $K$ be a field such that the additive group of $K$ is finitely generated as a group. Prove that $K$ is finite.

Let $F = \left\{ a+b\sqrt[3]{2}+c\sqrt[3]{4} \mid a,b,c \in \mathbb {Q}\right\}$. Prove that $F$ is a field and each element in $F$ has a unique representation as $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ in $F$, $c \in \mathbb {Q}$. Find $(1-\sqrt[3]{2})^{-1}$.

Let $F$ be a finite field. Give a complete proof of the fact that the number of elements of $F$ is of the form $p^r$, where $p \geq 2$ is a prime number and $r$ is an integer $\geq 1$.

Let $K$ be a field and $L \subset K$ a subfield containing $\left\{ a^2 \mid a \in K\right\}$. Prove that if the characteristic of $K$ is not 2 then $L = K$ and that the same conclusion holds for finite fields $K$ of characteristic 2, but not in general for fields of characteristic 2.

Let $F$ be a field of characteristic $p > 0$, $p \neq 3$. If $a$ is a zero of the polynomial $f(x) = x^p - x + 3$ in an extension field of $F$, show that $f(x)$ has $p$ distinct zeros in the field $F(a)$.

Let $K$ be the field $\mathbb {Q}(\sqrt[10]{2})$. Prove that $K$ has degree 10 over $\mathbb {Q}$, and that the group of automorphisms of $K$ has order 2.

Let $f(x) = x^5-8x^3+9x-3$ and $g(x) = x^4-5x^2-6x+3$. Prove that there is an integer $d$ such that the polynomials $f(x)$ and $g(x)$ have a common root in the field $\mathbb {Q}(\sqrt{d})$. What is $d$?

Let $F$ be a field containing $\mathbb {Q}$ such that $[F:\mathbb {Q}] = 2$. Prove that there exists a unique integer $m$ such that $m$ has no multiple prime factors and $F$ is isomorphic to $\mathbb {Q}(\sqrt{m})$.

Exhibit infinitely many pairwise nonisomorphic quadratic extensions of $\mathbb {Q}$ and show they are pairwise nonisomorphic.

Let $\mathbb {Q}$ be the field of rational numbers. For $\theta$ a real, let $F_\theta = \mathbb {Q}(\sin \theta )$ and $E_\theta = \mathbb {Q}(\sin \frac{\theta }{3})$. Show that $E_\theta$ is an extension field of $F_\theta$, and determine all possibilities for $\dim_{F_\theta } E_\theta$.

Show that the field $\mathbb {Q}(t_1, \ldots , t_n)$ of rational functions in $n$ variables over the rational numbers is isomorphic to a subfield of $\mathbb {R}$.

Let $f(x) \in \mathbb {Q}^{x}$ be an irreducible polynomial of degree $n \geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha \in L$ be a zero of $f$. Given that $[L:\mathbb {Q}] = n!$, prove that $\mathbb {Q}(\alpha^4) = \mathbb {Q}(\alpha )$.

Let $K$ be a field. For what pairs of positive integers $(a,b)$ is the subring $K[t^a, t^b]$ of $K[t]$ a unique factorization domain?

Let $F$ be a finite field of cardinality $p^n$, with $p$ prime and $n > 0$, and let $G$ be the group of invertible $2 \times 2$ matrices with coefficients in $F$.

Let $p$ be a prime and $k, n$ positive integers. Prove that the group $GL_n(\mathbf{F}_p)$ (the field of $p$ elements) of invertible $n \times n$ matrices over $\mathbf{F}_p$ contains an element of order $p^k$ if and only if $p^{k-1} < n$.

Let $F$ be a field, and let $G$ be the group of $2 \times 2$ upper-triangular matrices over $F$ of determinant 1.

Let $p$ be an odd prime and $\mathbf{F}_p$ the field of $p$ elements. How many elements of $\mathbf{F}_p$ have square roots in $\mathbf{F}_p$? How many have cube roots in $\mathbf{F}_p$?

Let $F$ be a finite field with $q$ elements. Say that a function $f : F \rightarrow F$ is a polynomial function if there are elements $a_0, a_1, \ldots , a_n$ of $F$ such that $f(x) = a_0+a_1 x+\cdots +a_n x^n$ for all $x \in F$. How many polynomial functions are there?

Let $F$ be a finite field, and suppose that the subfield of $F$ generated by $\left\{ x^3 \mid x \in F\right\}$ is different from $F$. Show that $F$ has cardinality 4.

Suppose that $A$ is a commutative algebra with identity over $\mathbb {C}$ (i.e., $A$ is a commutative ring containing $\mathbb {C}$ as a subring with identity). Suppose further that $a^2 \neq 0$ for all nonzero elements $a \in A$. Show that if the dimension of $A$ as a vector space over $\mathbb {C}$ is finite and at least two, then the equation $a^2 = a$ is satisfied by at least three distinct elements $a \in A$.

Let $A$ be a commutative ring with 1. Prove that the group $A^{*}$ of units of $A$ does not have exactly 5 elements.

Suppose $K$ is a field and $R$ is a nonzero $K$-algebra generated by two elements $a$ and $b$ which satisfy $a^2 = b^2 = 0$ and $(a+b)^2 = 1$. Show $R$ is isomorphic to $M_2(K)$ (the algebra of $2 \times 2$ matrices over $K$).