Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple extension of F and, indeed, E = F(@) for every a e E not in F.

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10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Section 31 number 23
Theory
22. Eet (a bijeCwhere a, beR and b # 0. Show that C = Ra + bi).
23. Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple extension of
F and, indeed, E = F(@) for every a e E not in F.
24. Prove that x² – 3 is irreducible over Q/2).
nhtoin bu cuccessively adioining to a field F a square root of an element
Transcribed Image Text:Theory 22. Eet (a bijeCwhere a, beR and b # 0. Show that C = Ra + bi). 23. Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple extension of F and, indeed, E = F(@) for every a e E not in F. 24. Prove that x² – 3 is irreducible over Q/2). nhtoin bu cuccessively adioining to a field F a square root of an element
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