930 Extra Problems Field Theory

Field Theory

General Algebra

• Show that any finite integral domain is a field.
• Show that every field is simple.
• Show that any field morphism is either 0 or injective.
• Show that if $L/F$ and $\alpha$ is algebraic over both $F$ and $L$, then the minimal polynomial of $\alpha$ over $L$ divides the minimal polynomial over $F$.
• Prove that if $R$ is an integral domain, then $R[t]$ is again an integral domain.
• Show that $ff(R[t]) = ff(R)(t)$.
• Show that $[{\mathbf{Q}}(\sqrt 2 + \sqrt 3) : {\mathbf{Q}}] = 4$.
  • Show that ${\mathbf{Q}}(\sqrt 2 + \sqrt 3) = {\mathbf{Q}}(\sqrt 2 - \sqrt 3) = {\mathbf{Q}}(\sqrt 2, \sqrt 3)$.
• Show that the splitting field of $f(x) = x^3-2$ is ${\mathbf{Q}}(\sqrt[3]{2}, \zeta_2)$.

Extensions?

• What is $[{\mathbf{Q}}(\sqrt 2 + \sqrt 3): {\mathbf{Q}}]$?
• What is $[{\mathbf{Q}}(2^{3\over 2}) : {\mathbf{Q}}]$?
• Show that if $p\in {\mathbf{Q}}[x]$ and $r\in {\mathbf{Q}}$ is a rational root, then in fact $r\in {\mathbf{Z}}$.
• If $\left\{{\alpha_i}\right\}_{i=1}^n \subset F$ are algebraic over $K$, show that $K[\alpha_1, \cdots, \alpha_n] = K(\alpha_1, \cdots, \alpha_n)$.
• Show that $\alpha/F$ is algebraic $\iff F(\alpha)/F$ is a finite extension.
• Show that every finite field extension is algebraic.
• Show that if $\alpha, \beta$ are algebraic over $F$, then $\alpha\pm \beta, \alpha\beta^{\pm 1}$ are all algebraic over $F$.
• Show that if $L/K/F$ with $K/F$ algebraic and $L/K$ algebraic then $L$ is algebraic.

Special Polynomials

• Show that a field with $p^n$ elements has exactly one subfield of size $p^d$ for every $d$ dividing $n$.
• Show that $x^{p^n} - x = \prod f_i(x)$ over all irreducible monic $f_i$ of degree $d$ dividing $n$.
• Show that $x^{p^d} - x \divides x^{p^n} - x \iff d \divides n$
• Prove that $x^{p^n}-x$ is the product of all monic irreducible polynomials in ${ \mathbf{F} }_p[x]$ with degree dividing $n$.
• Prove that an irreducible $\pi(x)\in { \mathbf{F} }_p[x]$ divides $x^{p^n}-x \iff \deg \pi(x)$ divides $n$.