I am studying Galois theory. Here I found a theorem known as Dedekind-Artin Theorem in which I stumbled upon at the far end of the proof. Let me discuss the theorem and then I will ask for clarification of my query.
Dedekind-Artin Theorem $:$
Let $L|K$ be a finite field extension. Then $\#\ \text{Gal}\ (L|K) \leq [L:K].$
Before going to the proof let me discuss Artin's lemma. Let $(M,.)$ be a monoid and $L$ be a field. Let $\chi (M,L)$ denote the set of all monoid homomorphisms from $M$ to $L.$ The elements of $\chi (M,L)$ are called the characters of $M$ with values in $L.$Clearly $\chi (M,L) \subseteq L^M =$ the set of all maps from $L$ to $L.$ It is clear that $L^M$ is a $L$-vector space w.r.t. componentwise addition and componentwise scalar multiplication. Then Artin's lemma states as follows $:$
$\chi (M,L)$ is a linearly independent (over $L$) subset of $L^M.$
Proof $:$ If not we get a non-trivial relation with coefficients in $L$ as follows $:$
$$z_1\chi_1 + z_2\chi_2 + \cdots + z_n\chi_n = 0.\ \ \ \ \ \ (1)$$ where $\chi_1,\chi_2,\cdots,\chi_n \in \chi (M,L)$ and $z_1,z_2,\cdots,z_n \in L.$ Le us assume that $n$ to be minimal. Hence we have $z_i \neq 0$ for $i=1,2\cdots ,n.$ and $\chi_i \neq \chi_j$ for $i \neq j$ where $i=1,2,\cdots ,n;\ j=1,2,\cdots,n.$ So in particular $\chi_1 \neq \chi_2.$ So $\exists\ x \in M$ such that $\chi_1 (x) \neq \chi_2 (x).$ Let us take $y \in M$ arbitrarily. Then $xy \in M,$ since $(M,.)$ is a monoid. So we get \begin{align*} (z_1\chi_1 + z_2\chi_2 + \cdots + z_n\chi_n)(xy) &= 0\\ \implies z_1\chi_1(xy) + z_2\chi_2(xy) + \cdots + z_n\chi_n(xy) &= 0\\ \implies z_1\chi_1(x) \chi_1(y) + z_2\chi_2(x) \chi_2(y) + \cdots + z_n\chi_n(x) \chi_n(y) &= 0.\end{align*} Since $y$ was arbitrarily taken so we have $$z_1\chi_1(x) \chi_1 + z_2\chi_2(x) \chi_2 + \cdots + z_n\chi_n(x) \chi_n = 0.\ \ \ \ \ \ (2)$$Multiplying $(1)$ by $\chi_1(x)$ and then substracting it from $(2)$ we get $$z_2(\chi_2 -\chi_1)(x)\chi_1+\cdots+z_n (\chi_n-\chi_1)(x)\chi_n=0.$$ Since $z_2 \neq 0$ and $\chi_1(x) \neq \chi_2 (x)$ in $L$ so we have $z_2(\chi_2-\chi_1)(x) \neq 0$ since $L$ contains no divisor of zero. So we get a non-trivial relation of the elements of $\chi (M,L)$ with coefficients in $L$ whose length is $\leq n-1.$ So it contradicts the minimality of $n$ and hence the lemma follows.
Now let us prove Dedekind-Artin Theorem with the help of Artin's lemma.
Choose $M=L.$ Then clearly $\text{Gal}\ (L|K) \subseteq \text {End}_K(L) \subseteq L^L,$ where $\text{End}_K(L)$ is the set of all $K$-linear maps on $L.$ Now we know that $\text {End}_K(L)$ is a $K$-vector space. Now let $z \in L$ and $f \in \text {End}_K(L).$ Define $(zf) (x) = z f(x),$ for all $x \in L.$ Then w.r.t. that scalar multiplication $\text {End}_K(L)$ becomes an $L$-vector space. We know that $\text {Dim}_K\ (\text {End}_K(L)) = [L:K]^2.$ Also we know that $$\text {Dim}_K\ (\text {End}_K(L)) = \text {Dim}_L\ (\text {End}_K(L))\ [L:K].$$ Therefore $\text {Dim}_L\ (\text {End}_K(L)) = [L:K].$ Now $\chi (L,L) \subseteq \text {End}_K (L)$ and since $\chi (L,L)$ is linearly independent over $L$ we must have $$\#\ \chi (L,L) \leq \text {Dim}_L\ (\text {End}_K(L)) = [L:K].$$ But it is clear that $\text {Gal} (L|K) \subseteq \chi (L,L).$ So we have $$\#\ \text {Gal} (L|K) \leq \#\ \chi (L,L) \leq [L:K].$$
QED
Here I don't understand why does $\chi (L,L) \subseteq \text{End}_{K} (L)$? Let $f \in \chi (L,L).$ Then two properties will be satisfied \begin{align*} (1)\ f(xy)&=f(x)f(y),\ \text {for all}\ x,y \in L. \\ (2)\ f(1_L) &= 1_L. \end{align*} But how do these two properties guarantee the $K$-linearity of $f$? Any help regarding this will be highly appreciated.
Thank you very much.
You're right, the end of the proof seems flawed (or mistyped), and $\chi(L,L)\subseteq\mathrm{End}_K(L)$ indeed might not hold. But we can fix it.
Still, we get that the elements of $\mathrm{Gal}(L\mid K)\subseteq\chi(L,L)\subseteq L^L$ are $L$-linearly independent.
And, we also have $\mathrm{Gal}(L\mid K)\subseteq \mathrm{End}_K(L)$, and they are still independent in $\mathrm{End}_K(L)$ which is a subspace of $L^L$ over $L$, with the given scalar multiplication.