3. Suppose R and S are fields, with Ra subring of S. (We say R C S is a field extension.) For a polynomial p = co+ ar++ Cr" in R(r), and an element a of S, write p(a) for the element co +eja ++c,a" of S. We say a is a root of p in S if p(a) = 0. For a e S, let R(a) = {p(a) |pe Rz}}. Clearly R(a] is a subring of S. Prove: if a is a root of some nonzero polynomial pE R2), then Raj is a field, and otherwise Rla] is isomorphic to R(z). Hint (for the first case): Show Ra) is isomorphic to a quotient of a commutative ring by a maximal ideal. (Since R is a domain, if a is a root of pq, then a is a root of p or of q; use item 2.)

3. Suppose R and S are fields, with Ra subring of S. (We say R C S is a field extension.) For a polynomial p = co+ ar++ Cr" in R(r), and an element a of S, write p(a) for the element co +eja ++c,a" of S. We say a is a root of p in S if p(a) = 0. For a e S, let R(a) = {p(a) |pe Rz}}. Clearly R(a] is a subring of S. Prove: if a is a root of some nonzero polynomial pE R2), then Raj is a field, and otherwise Rla] is isomorphic to R(z). Hint (for the first case): Show Ra) is isomorphic to a quotient of a commutative ring by a maximal ideal. (Since R is a domain, if a is a root of pq, then a is a root of p or of q; use item 2.)

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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