Do as asked.a. Give the smallest extension of Q that contains all the zeros of x4 – 2x2 – 15.b. Let m, n be positive integers. Show that the real numbers em and e are algebraic overQ(e"). [Here, e is Euler's number.]c. Let a =V7+4/3. Show that [Q(a) : Q] = 2.d. Let p be a prime. Give all the subfields of F,20 and sketch its subfield lattice.

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Answered: a. Give the smallest extension of Q…

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a. Give the smallest extension of Q that contains all the zeros of x4 - 2x2 – 15.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 37E

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Question

Do as asked.
a. Give the smallest extension of Q that contains all the zeros of x4 – 2x2 – 15.
b. Let m, n be positive integers. Show that the real numbers em and e are algebraic over
Q(e"). [Here, e is Euler's number.]
c. Let a =
V7+4/3. Show that [Q(a) : Q] = 2.
d. Let p be a prime. Give all the subfields of F,20 and sketch its subfield lattice.

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