(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?

(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

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Question

Recall that, given fields K ⊂ L and an element u ∈ L \ K, we write
K(u) = {k₀ + k₁ u + k₂u² + · · · + k_nuⁿ : k_i ∈ 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?

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