| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
35 |
IB |
2019 |
Groups, Rings and Modules |
|
Paper 2, Section II, G |
2019 |
(a) Let $k$ be a field and let $f(X)$ be an irreducible polynomial of degree $d>0$ over $k$. Prove that there exists a field $F$ containing $k$ as a subfield such that
$$f(X)=(X-\alpha) g(X)$$
where $\alpha \in F$ and $g(X) \in F[X]$. State carefully any results that you use.
(b) Let $k$ be a field and let $f(X)$ be a monic polynomial of degree $d>0$ over $k$, which is not necessarily irreducible. Prove that there exists a field $F$ containing $k$ as a subfield such that
$$f(X)=\prod_{i=1}^{d}\left(X-\alpha_{i}\right)$$
where $\alpha_{i} \in F$.
(c) Let $k=\mathbb{Z} /(p)$ for $p$ a prime, and let $f(X)=X^{p^{n}}-X$ for $n \geqslant 1$ an integer. For $F$ as in part (b), let $K$ be the set of roots of $f(X)$ in $F$. Prove that $K$ is a field.