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 \(f(x) = x^32\) 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 \(n\).