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Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its determinant. Now one can view $T$ as an $F$-linear operator on $V$ by viewing $V$ as an $F$-vector space. Let $\det_F(T)$ be the determinant of $T$ viewed as an $F$-linear operator. Is it true that

$\det_F(T)=N_{E/F}(\det(T))$

where $N_{E/F}$ is the norm from $E$ to $F$?

If so, is there any reference or simple proof?

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    $\begingroup$ As with many things, this follows from Theorem 1 in John R. Silvester, Determinants of Block Matrices, The Mathematical Gazette, Vol. 84, No. 501 (Nov., 2000), pp. 460--467. In fact, the matrix that represents $T$ as an $F$-linear operator on $V$ is a block matrix, whose blocks are elements of $F$ written as square matrices over $E$; clearly these blocks commute. Separability of $E/F$ is unnecessary. $\endgroup$
    – darij grinberg
    Commented Sep 6, 2018 at 4:26
  • $\begingroup$ This is a special case of Bourbaki, Algèbre III.9.4 Proposition 6. $\endgroup$
    – Fred Rohrer
    Commented Sep 6, 2018 at 11:03
  • $\begingroup$ @darijgrinberg it is not true that these blocks commute. For instance if $A+iB$ is a complex $n\times n$ matrix, then the matrix representing it over $\mathbb{R}$ is the $2\times 2$ block matrix with blocks $A ,-B ,B ,A$, but $A,B$ don't commute. I don't see how theorem 1 implies the original question. $\endgroup$
    – Espace' etale
    Commented Mar 31, 2020 at 13:12
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    $\begingroup$ @Espace'etale: You want to represent it as an $n\times n$ block matrix made out of commuting $2 \times 2$-blocks, not as a $2\times 2$ block matrix made out of commuting $n \times n$-blocks. Each $2 \times 2$-block will have the form $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ for two real numbers $a$ and $b$; but all such blocks commute. $\endgroup$
    – darij grinberg
    Commented Mar 31, 2020 at 13:18
  • $\begingroup$ Very good, wonderful! Infact the blocks will in general be members of an embedding of $E$ inside the ring of matrices over $E$ of size $[E:F]$, right? $\endgroup$
    – Espace' etale
    Commented Mar 31, 2020 at 16:01

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