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).}

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305 -Extensions by radicals (2)

March 5, 2009

2. Extensions by radicals


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.

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