# 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 $${\mathbf{Q}}(\sqrt{2}, \zeta_2)$$.

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 $$n$$.