Suppose that K is the splitting field of some polynomial over a fieldF of characteristic 0. If [K:F] = p2q, where p and q are distinct primes,show that K has subfields L1, L2, and L3 such that [K:L1] = p, [K:L2]= p2, and [K:L3] = q.

Suppose that K is the splitting field of some polynomial over a fieldF of characteristic 0. If [K:F] = p2q, where p and q are distinct primes,show that K has subfields L1, L2, and L3 such that [K:L1] = p, [K:L2]= p2, and [K:L3] = q.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 18E

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Question

Suppose that K is the splitting field of some polynomial over a field
F of characteristic 0. If [K:F] = p²q, where p and q are distinct primes,
show that K has subfields L₁, L₂, and L₃ such that [K:L₁] = p, [K:L₂]
= p², and [K:L3] = q.

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