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Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, ab, ab-¹ have degree at most mn over K.

Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, ab, ab-¹ have degree at most mn over K.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Field Extension 

P, pa prime. Prove that if c E F, c# K, then F = K (c). Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, a - b, ab-1 have degree at most mn over K.
Transcribed Image Text:P, pa prime. Prove that if c E F, c# K, then F = K (c). Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, a - b, ab-1 have degree at most mn over K.
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