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New Proofs for two Theorems of Capelli

Published online by Cambridge University Press:  20 November 2018

Elizabeth Rowlinson
Elizabeth Rowlinson*
Affiliation:
McGill University
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Extract

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The following two theorems are due to Capelli.

THEOREM 1. Let g(x) and h(x) be polynomials over a field R of characteristic 0; let f(x) = g(h(x)). Then f(x) is irreducible over R if and only if

  1. (i) g(x) is irreducible over R

and

  1. (ii) h(x) - β is irreducible over R(β), where β is a root of g(x).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 3 , September 1964 , pp. 431 - 433
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Tschebotarőw, N. and Schwerdtfeger, H., Grundzűge der Galois' schen Theorie, Noordhoff, Groningen (1950).Google Scholar
2. Van der Waerden, B. L., Modern Algebra, Vol. I, Ungar, (1953).Google Scholar