## 305 -Extensions by radicals (3)

March 8, 2009

3. Examples

Last lecture we defined what it means that a polynomial with coefficients in a field ${{\mathbb F}}$ is solvable by radicals over ${{\mathbb F}}$. Namely, there is a tower of extensions

$\displaystyle {\mathbb F}(t_1,\dots,t_k):{\mathbb F}(t_1,\dots,t_{k-1}):\dots:{\mathbb F}(t_1):{\mathbb F}$

where for each ${j,}$ ${1\le j\le k,}$ there is a positive integer ${m_j}$ such that ${t_j^{m_j}\in{\mathbb F}(t_1,\dots,t_{j-1}),}$ and such that ${{\mathbb F}^{p(x)}\subseteq{\mathbb F}(t_1,\dots,t_k).}$

Last lecture we defined ${{\mathbb F}^{p(x)}}$ where ${{\mathbb F}}$ is a subfield of a field ${{\mathbb K},}$ all the roots of the polynomial ${p(x)}$ are in ${{\mathbb K},}$ and all the coefficients of ${p(x)}$ are in ${{\mathbb F}.}$ Namely, if ${r_1,\dots,r_n}$ are the roots of ${p,}$ then ${{\mathbb F}^{p(x)}={\mathbb F}(r_1,\dots,r_n),}$ the field generated by ${r_1,\dots,r_n}$ over ${{\mathbb F}.}$
The typical examples we will consider are those where ${{\mathbb F}={\mathbb Q},}$ ${{\mathbb K}={\mathbb C},}$ and the coefficients of ${p(x)}$ are rational or in fact, integers.