Question

Let K is Galois over F and $\mathit{G}\mathit{a}{\mathit{l}}_{\mathbf{F}}\mathit{K}$ is an abelian group of order 10 how many intermediate field does the extension have and what are their dimensions over F .

Short answer

There are 4 intermediate field and their dimension are$\mathbf{<}\mathbf{1}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{2}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{5}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{10}\mathbf{>}$ .

Step-by-step solution

Definition of the abelian group.

An abelian group is a commutative group and satisfy the condition for all $\mathit{a}\mathbf{,}\mathit{b}$ in the field extension isa-b=b.a .

Step-2: Showing that there are 4 intermediate field and there dimensions are <1>,<2>,<5>,<10> .

Consider the Galois field $\mathit{K}$ and Galois group of the extension is abelian group of the order of 10. Now compute the number of the intermediate field which has existed in the extension field and also the dimension over the field $\mathit{F}$.

Now there is the only abelian group of order 10 up to isomorphism is ${\mathit{Z}}_{\mathbf{10}}$ it has four subgroups that are localid="1657962012982" $\mathbf{<}\mathbf{1}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{2}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{5}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{10}\mathbf{>}$.

Those are the dimension of the four intermediate fields of the extension field. Since $\mathit{K}$ is Galois field over $\mathit{F}$ these are only intermediate field of the extension by using Galois theory.

Hence there are 4 intermediate fields and its dimensions are $\mathbf{<}\mathbf{1}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{2}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{5}\mathbf{>}\mathbf{,}\mathbf{<}\mathbf{10}\mathbf{>}$.