Defining the characteristic polynomial of an endomorphism through its properties instead of a formula The usual definition of the characteristic polynomial of an endomorphism $f$ of a finite dimensional $F$ vector space is $\chi_f(T)=\det(T\operatorname{id}-f)$. It's essentially just a formula. Meanwhile, the usual definition of the minimal polynomial is much more conceptual: the minimal polynomial $\mu_f$ of $f$ is the unique monic polynomial in $F[T]$ of least degree except the zero polynomial such that $\mu_f(f)=0$. (Or more abstractly the monic generator of $\ker\varphi_f$, where $\varphi_f:F[T]\to\operatorname{End}(V)$ is the evaluation homomorphism at $f$). This definition emphasizes the properties of the polynomial in question, instead of a way to calculate it.
Question: Is there a nice way to define the characteristic polynomial in a similar way? Bonus points if it makes the theory built on top easier (like a more straightforward proof of Cayley-Hamilton). Additional bonus points if the definition doesn't rely on the existence of an algebraic closure of $F$.
 A: Let $R$ be a PID (or more generally a Dedekind domain). Let $C$ be the category of finitely generated torsion modules over $R$. Then there is a unique way $F$ to assign to every element of $C$ an ideal of $R$ such that the following identities are satisfied:
$F(R/(a))=(a)$
$F(M\oplus N)=F(M) \cdot F(N)$
This $F$ is actually a special case of a more general construction, called the 0-th Fitting ideal denoted $\mathrm{Fitt}_0(M)$. Caley-Hamilton is a special case of the relation $\mathrm{Fitt}_0(M) \subset \mathrm{Ann}(M)$.
But we can also prove Caley-Hamilton from the characterization given above: Just use the structure theorem for finitely generated modules over a PID and note that if $M \cong R/(a_1) \oplus \dots R/(a_n)$, then we have $F(M)=(\prod a_i)$ and $\mathrm{Ann}(M)=(\mathrm{lcm}(a_1, \dots, a_n))$. As the least common multiple divides the product, we are done.
Anyway, if we have this assignment, we can associate to each finite-dimensional $k$-vector space with an endomorphism $f$ a finitely generated torsion $k[T]$-module and to that an ideal in $k[T]$, the characteristic polynomial is then the unique monic generator of that ideal.
To elaborate on how to get a $k[T]$-module from an endomorphism of a vector space, the construction is as follows: let $V$ be a vector space and let $f$ be an endomorphism of $V$. Then we can take $V$ as an abelian group and define the $k[T]$-module structure via the formula $(\sum a_i T^i)v=\sum a_i f^i(v)$, where $f^i$ denotes $f^{i}$ applied $i$ times. This yields a $k[T]$-module and if $V$ is finite-dimensional, the module is torsion, as indicated in the comments.
Note that this construction of associating a $k[T]$ module to a finite-dimensional $k$ vector space with an endomorphism as a way of studying the endomorphism has applications beyond characteristic polynomials and Caley-Hamilton. Indeed, via the structure theorem for finitely generated modules, it provides a conceptual and algebraic way of proving existence and uniqueness of the Jordan normal form and the Frobenius, which solves the problem of classifying endomorphisms up to similarity.
