Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= \begin{pmatrix} \alpha_1 & \sigma(\alpha_1) & \dots & \sigma^{n-1}(\alpha_1)\\ \vdots & \vdots &\ddots & \vdots\\ \alpha_n & \sigma(\alpha_n) &\dots & \sigma^{n-1}(\alpha_n)\\ \end{pmatrix} $$ represents the automorphism $\sigma\in \mathrm{Gal}(K/k)$ in the basis $\alpha_1,\dots,\alpha_n$. However I don't know how to show it. Can someone help me? Thanks!

Edit: As pointed out by Berci the statement that $M$ represents $\sigma$ is probably incorrect. In the notes I read this statement is used to show that $M$ is nonsingular. So here is the new question:

Question: Is there an alternative way of proving the nonsingularity of $M$ (or disproving it by finding a counter-example)?

P.S. I notice that in the context of algebraic number theory, the discriminant of an algebraic number field $K$ has a very similar form, being $(\det(M))^2$. So perhaps there is a connetion?

  • 2
    $\begingroup$ It must be some confusion somewhere.. At least, as in usual in linear algebra, the matrix $[\sigma]$ of $\sigma$ is a matrix with coefficients in $k$, and its columns are $(\sigma(\alpha_1),\sigma(\alpha_2),\dots,\sigma(\alpha_n))$, all these coordinated in the basis $\alpha_1,..,\alpha_n$. $\endgroup$
    – Berci
    Jun 17, 2013 at 1:43
  • $\begingroup$ Thanks! I've edit the post and ask a new question. $\endgroup$
    – Zeyu
    Jun 17, 2013 at 11:25
  • $\begingroup$ What may be of interest is the Skolem-Noether theorem. One of its consequences is that if you represent the field $K$ as $n\times n$ matrices with entries in $k$, call this mappin $\rho:K\to M_n(k)$, then to each automorphism $\sigma\in Gal(K/k)$ there exists a (non-unique) matrix $A_\sigma\in M_n(k)$ such that $$\rho(\sigma(x))=A_\sigma \rho(x) A_\sigma^{-1}.$$ In other words, the elements of the Galois group become conjugations in the general linear group. $\endgroup$ Jun 17, 2013 at 11:52
  • $\begingroup$ "I read somewhere that the following matrix ..." - do you remember where? $\endgroup$ May 6, 2017 at 20:08


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