I am aware of the characteristics polynomial of a linear operator on a finite dimensional $K$-vector space $V.$ Let $V$ be a $K$-vector space of dimension $n$ and $T:V \longrightarrow V$ be a linear operator on $V.$ Then the characteristics polynomial $\chi_T$ of $T$ is defined as $$\chi_T := \det (XI_n-M)$$ where $M$ is the matrix representation of $T$ w.r.t. some fixed $K$-basis $B$ (say) of $V.$ Clearly $\chi_T \in K[X].$
I am studying Galois theory from NPTEL lecture series. Here I found an assertion made by the instructor which states that " If $L|K$ be a finite Galois extension then for any $y \in L$ the characteristics polynomial $\chi_y$ of $y$ over $K$ is of the form $$\chi_y = \prod_{\sigma \in \text {Gal}\ (L|K)} (X-\sigma (y)).\text{''}$$ But I don't know what's the definition of the charateristics polynomial of some element of the extended field over the base field if the field extension is finite. Is there any linear operator related argument present here too? Because where $L|K$ be a finite field extensioin then we can always think of $L$ as a finite dimensional $K$-vector space. Let $y \in L.$ So there may exist a $K$-linear operator $T$ such that characteristics polynomial $\chi_T$ of $T$ matches with that of the characteristics polynomial $\chi_y$ of $y$ over $K$ and also same thing happens for minimal polynomials. If so then I can relate this two concepts quite easily. Like in linear algebra we know that the minimal polynomial of a linear operator shares same prime factors with the characteristics polynomial. So the concept of characteristics and minimal polynomial in linear algebra matches with the finite field extensions then we can certainly say that the characteristics polynomial of some element is a power of it's minimal polynomial because minimal polynomial of some element of the extended field over the base field is a prime polynomial over the base field for finite field extensions.
Can anybody please give me any sughgestion regarding this? Thank you very much.
My knowledge on this subject is a bit rough, but here's what I think is going on.
Let $L/K$ be a finite field extension and let $\alpha \in L$. Consider the $K$-linear map $m_\alpha\colon L\to L: x\mapsto \alpha x$. The characteristic polynomial of $m_\alpha$ is the defined to be the characteristic polynomial of the element $\alpha\in L$.
In particular, if $\alpha\in K$, then $m_\alpha=cId_L$ and thus the characteristic polynomial is just $(X-\alpha)^n$ where $n=\dim_K(L)$ which is consistent with the formula you give for Galois extensions.
So the only thing going on here to extend the usual definition of characteristic polynomial to an element, is to view said element as an operator acting by multiplication by said element.