MHB Showing Linearity of $\varphi$ for $K(a)$

[mathmari]

Hey!

Let $K \leq K(a)$ a field extension with $[K(a):K]=n$.

$K(a)$ is a vector space over $K$.

How can I show that the map $\varphi : K(a) \rightarrow K(a)$, with $\varphi(e)=ae$, is a $K-$linear map??

 

[Euge]

Hey!

Let $K \leq K(a)$ a field extension with $[K(a):K]=n$.

$K(a)$ is a vector space over $K$.

How can I show that the map $\varphi : K(a) \rightarrow K(a)$, with $\varphi(e)=ae$, is a $K-$linear map??

Hi mathmari,

To show that $\varphi$ is $K$-linear, you must verify

$$ \varphi(te + e') = t\varphi(e) + \varphi(e') \quad \text{for all} \quad t\in K, \, e, e'\in K(a).$$

 

[mathmari]

Hi mathmari,

To show that $\varphi$ is $K$-linear, you must verify

$$ \varphi(te + e') = t\varphi(e) + \varphi(e') \quad \text{for all} \quad t\in K, \, e, e'\in K(a).$$

I understand! Thank you! (Sun)

I have also an other question...

I am also asked to show that $a$ is a root of the characteristic polynomial of $\varphi$.

Which is the characteristic polynomial of $\varphi$?? (Wondering)

 

[Euge]

I understand! Thank you! (Sun)

I have also an other question...

I am also asked to show that $a$ is a root of the characteristic polynomial of $\varphi$.

Which is the characteristic polynomial of $\varphi$?? (Wondering)

It's $p(x) = \text{det}(x1_{K(a)} - \varphi)$. The map $x1_{K(a)} - \varphi$ sends $e \in K(a)$ to $xe - ae$.

 

[mathmari]

It's $p(x) = \text{det}(x1_{K(a)} - \varphi)$. The map $x1_{K(a)} - \varphi$ sends $e \in K(a)$ to $xe - ae$.

Could you explain me what $x1$ is ?? (Wondering)

 

[Euge]

Could you explain me what $x1$ is ?? (Wondering)

The map $x1_{K(a)}$ sends $e\in K(a)$ to $xe$.