Question

If$\mathit{\sigma }$ is an F-automorphism of K, show that${\mathit{\sigma }}^{\mathbf{-}\mathbf{1}}$ is also an F-automorphism of K.

Short answer

It is proved that${\mathit{\sigma }}^{\mathbf{-}\mathbf{1}}$ is an F-automorphism of K.

Step-by-step solution

Definition of  F-automorphism

An extension field K of F , an automorphism of K is an isomorphism$\mathit{\sigma }\mathbf{:}\mathit{K}\mathbf{\to }\mathit{K}$ isfixedF. Since$\mathit{\sigma }\left(c\right)\mathbf{=}\mathit{c}$ for every $\mathit{c}\mathbf{\in }\mathit{F}$.

Showing that σ-1 is automorphism of K.

If is an F-automorphism of K, then it is also the isomorphism and fulfil the condition of one to one and onto. The one to one and onto condition has a property of addition and scaling that is given below:

$$\begin{gathered} \mathbf{f}\left(a+b\right)\mathbf{=}\mathbf{f}\left(a\right)\mathbf{+}\mathbf{f}\left(b\right)\mathbf{x}\in \left(a,b\right) \\ \mathbf{f}\left(\mathrm{cV}\right)\mathbf{=}\mathbf{cf}\left(V\right)\mathrm{in}\mathrm{same}\mathrm{field}\mathbf{V}\mathbf{\to }\mathbf{V} \end{gathered}$$

If $\mathit{\sigma }$ is an automorphism, then $\mathit{F}\mathbf{\subset }\mathit{K}$. Here, F is the content of large field K. Now, let us choose two elements x and y that satisfies the isomorphism conditions.

Now, conclude that in same field ${\mathit{\sigma }}^{\mathbf{-}\mathbf{1}}$is also the isomorphism in $\mathit{F}\mathbf{\subset }\mathit{K}$.

The properties are:

role="math" localid="1657956335486" $$\begin{gathered} {\mathbf{\sigma }}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{=}{\mathbf{\sigma }}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}{\mathbf{\sigma }}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{K}\mathbf{\in }\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{\sigma }}^{\mathbf{-}\mathbf{1}}\left(\mathrm{cx}\right)\mathbf{=}{\mathbf{c\sigma }}^{\mathbf{-}\mathbf{1}}\left(x\right)\mathrm{in}\mathrm{same}\mathrm{field}\mathbf{}\mathbf{V}\mathbf{\to }\mathbf{V} \end{gathered}$$

Since, the element x is the content of the subfield F . Hence,${\mathit{\sigma }}^{\mathbf{-}\mathbf{1}}$ is also the automorphism in the field $\mathit{F}\mathbf{\subset }\mathit{K}$.