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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.2: Integral Domains And Fields
Problem 3E: Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as...
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb1306d9-745d-43d2-9d3a-24c5b601ff6f%2F4b5809a8-dc66-49a7-a140-a98365d7214b%2Fuhlnntk_processed.jpeg&w=3840&q=75)
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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