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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the order and degree of solutions to pure equations


Author: Lawrence J. Risman
Journal: Proc. Amer. Math. Soc. 55 (1976), 261-266
DOI: https://doi.org/10.1090/S0002-9939-1976-0396508-4
MathSciNet review: 0396508
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Abstract: Let $ K$ be a field. Let $ \theta $ be an element of a field extension of $ K$. The order of $ \theta $ over $ K$ is the smallest positive integer $ m$ such that $ {\theta ^m}$ lies in $ K$, or $ \infty $. We compare the order $ m$ of $ \theta $ to the degree $ h$ of $ \theta $ over $ K$. Clearly $ h \leqslant m$.

Theorem. Let $ K$ be a field. Let $ \theta $ be an element of degree $ h$ and order $ m$ over $ K$. Let $ p$ be a prime. Let $ {p^e}$ be the maximum power of $ p$ dividing $ h$, and suppose $ {p^s}$ divides $ m$.

(1) If the characteristic of $ K$ is $ p$, then $ s \leqslant e$.

(2) If $ s > e$ and $ p$ is odd, then $ K(\theta )$ contains a primitive $ p$th root of unity $ u$ not in $ K$. Moreover $ K(u)$ contains a primitive $ {p^{s - e}}$ root of unity.

(3) If $ s > e$ and $ p = 2$, then $ - 1$ is not a square in $ K$ and $ K(\theta )$ contains $ i = \sqrt { - 1} $. Moreover $ - 1$ is a $ {2^{s - e}}$ power in $ K(i)$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0396508-4
Article copyright: © Copyright 1976 American Mathematical Society