On the order and degree of solutions to pure equations

Author:
Lawrence J. Risman

Journal:
Proc. Amer. Math. Soc. **55** (1976), 261-266

MathSciNet review:
0396508

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Abstract | References | Additional Information

Abstract: Let be a field. Let be an element of a field extension of . The order of over is the smallest positive integer such that lies in , or . We compare the order of to the degree of over . Clearly .

Theorem. *Let* *be a field. Let* *be an element of degree* *and order* *over* . *Let* *be a prime. Let* *be the maximum power of* *dividing* , *and suppose* *divides* .

(1) *If the characteristic of* *is* , *then* .

(2) *If* *and* *is odd, then* *contains a primitive* *th root of unity* *not in* . *Moreover* *contains a primitive* *root of unity*.

(3) *If* *and* , *then* *is not a square in* *and* *contains* . *Moreover* *is a* *power in* .

**[1]**Israel N. Herstein, Claudio Procesi, and Murray Schacher,*Algebraic valued functions on noncommutative rings*, J. Algebra**36**(1975), no. 1, 128–150. MR**0374185****[2]**Irving Kaplansky,*Fields and rings*, The University of Chicago Press, Chicago, Ill.-London, 1969. MR**0269449****[3]**Serge Lang,*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234****[4]**Lawrence Risman,*On the multinomial degree of an element and solutions to pure equations*, Technion Preprint Series No. MT-234, 1975.

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0396508-4

Article copyright:
© Copyright 1976
American Mathematical Society