(a) Verify that Q(√3 ) is a subfield of R. (b) Show that Q(√3 ) is isomorphic to the quotient Q[x] / (x2 − 3) . (c) Using what you’ve learned from parts (a) and (b), describe the quotient ring F[x] / (x − a) when c ∈ F . Is this isomorphic to anything interesting?

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

Recall that, given fields K ⊂ L and an element u ∈ L \ K, we write
K(u) = {k0 + k1 u + k2u2 + · · · + knun : ki ∈ K, n ∈ N} for the smallest subfield of L containing K ∪ {u}.
(a) Verify that Q(√3 ) is a subfield of R.
(b) Show that Q(√3 ) is isomorphic to the quotient Q[x] / (x2 − 3) .
(c) Using what you’ve learned from parts (a) and (b), describe the quotient ring F[x] / (x − a) when c ∈ F . Is this isomorphic to anything interesting?

Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,