On the order and degree of solutions to pure equations
HTML articles powered by AMS MathViewer
- by Lawrence J. Risman PDF
- Proc. Amer. Math. Soc. 55 (1976), 261-266 Request permission
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 $K(i)$.References
- Israel N. Herstein, Claudio Procesi, and Murray Schacher, Algebraic valued functions on noncommutative rings, J. Algebra 36 (1975), no. 1, 128–150. MR 374185, DOI 10.1016/0021-8693(75)90160-X
- Irving Kaplansky, Fields and rings, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0269449
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234 Lawrence Risman, On the multinomial degree of an element and solutions to pure equations, Technion Preprint Series No. MT-234, 1975.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 261-266
- DOI: https://doi.org/10.1090/S0002-9939-1976-0396508-4
- MathSciNet review: 0396508