I had asked previously today about what the author of this proof meant by $\text{Tr}_K(\varphi : V \to V)$ in the statement of lemma 9.20.4. Now I'm trying to understand the proof itself, so I created this separate question. I will copy and paste the relevant information from the proof I linked:
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Let $L/K$ be a finite extension of fields. By Lemma 9.4.1 we can choose an isomorphism $L \cong K^{\oplus n}$ of K-modules. Of course $n=[L:K]$ is the degree of the field extension. Using this isomorphism we get for a $K$-algebra map
$$L \to \text{Mat}(n \times n,K),\alpha \to \text{matrix of multiplication by } \alpha $$
Thus given $\alpha \in L$ we can take the trace and the determinant of the corresponding matrix. Of course these quantities are independent of the choice of the basis chosen above. More canonically, simply thinking of $L$ as a finite dimensional $K$-vector space we have $\text{Trace}_K(\alpha:L→L)$ and the determinant $\det_K(\alpha:L→L)$.
Lemma 9.20.4: Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional vector space over $L$. Let $\varphi : L \to L$ be an $L$-linear map. Then
$$\text{Trace} _ {K}(\varphi : V \to V) = \text{Trace} _{L/K}(\text{Trace}_L(\varphi : V \to V))$$
$$ \det \nolimits _ K(\varphi : V \to V) = \text{Norm}_{L/K}(\det \nolimits _ L(\varphi : V \to V))$$
Proof: Choose an isomorphism $V=L^{\oplus n}$ so that $\varphi$ corresponds to an $n \times n$ matrix. In the case of traces, both sides of the formula are additive in $\varphi$. Hence we can assume that $\varphi$ corresponds to the matrix with exactly one nonzero entry in the $(i,j)$ spot. In this case a direct computation shows both sides are equal.
In the case of norms both sides are zero if $\varphi$ has a nonzero kernel. Hence we may assume $\varphi$ corresponds to an element of $\text{GL}_n(L)$. Both sides of the formula are multiplicative in $\varphi$. Since every element of $\text{GL}_n(L)$ is a product of elementary matrices we may assume that $\varphi$ either looks like
(because we may also permute the basis elements if we like). In both cases the formula is easy to verify by direct computation. $\blacksquare$
Lemma 9.20.5. Let $M/L/K$ be a tower of finite extensions of fields. Then
$\text{Trace} _ {M/K} = \text{Trace} _ {L/K} \circ \text{Trace} _ {M/L} \quad \text{and} \quad \text{Norm} _ {M/K} = \text{Norm} _ {L/K} \circ \text{Norm} _ {M/L}$
Proof. Think of $M$ as a vector space over $L$ and apply Lemma 9.20.4. $\blacksquare$
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Specifically, my question about this proof is about the "direct computation" the author mentions at the end of the first paragraph of lemma 9.20.4. How exactly is it performed? Also, he mentions $\varphi$ having exactly one nonzero entry. Does he refer to exactly one nonzero entry at a specific column? Because if he is referring to the whole matrix, then I didn't understand how he proves lemma 9.20.5, since the only $\alpha$ for which there's a column with only zero entries in the matrix of $x \to \alpha x$ is $\alpha=0$.