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 \(ff(R[t]) = ff(R)(t)\).
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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 \(n\).