Geodesic
Higher Derivations and the Jordan Canonical Form of the Companion Matrix
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 143-144

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The purpose of this note is to give a basis with respect to which the companion matrix of an equation (over a field of any characteristic) is in Jordan canonical form.Let k be a field. Define a k-linear map Di:k[X]→k[X] by where the integer is the binomial coefficient n!;/i!;(n —i)!;. We adopt the usual convention that if i>9 or j<0. Then D=(D 0, D 1D 2,...) is a higher derivation (see [1, p. 192]). 
Roberts, Leslie G. Higher Derivations and the Jordan Canonical Form of the Companion Matrix. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 143-144. doi: 10.4153/CMB-1972-027-6
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[1] 1. Jacobson, N., Lectures in abstract algebra, Vol. 3, Theory of fields and Galois theory. Van Nostrand, Princeton, N.J., 1964. Google Scholar

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