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

See similar textbooks

Similar questions

Suppose that b is a zero of f (x) = x4 + x + 1 in some extension field E of Z2. Write f (x) as a product of linear factors in E[x].
Show that the factor ring (P(x)) has a zero-divisor 1. Let P(x) = 3r3-14a2 + 29r 14. %3D
I need the answer as soon as possible
Determine with explanation which of the following polynomials is irreducible over the mentioned field. 1. x° +6x4+3x<+5x-2 over Q. 2. x4+x+1 over R.
Help with this please
5. Let 1 and g be polynomials of degree 3. Is it true that fog =go f? O A. No, it is only true for polynomials of degree 1. O B. Yes, always OC. It is true in some cases. D. No, never
20.* a Lei P(2) = 1+22 + 322. of P(2) are inside the unit disc. b. Show that the same conclusion applies to any polynomial of the form: ao +a1z+azz²+…+anz", with all a, real and 0 s ao sai s s an +nz"-1, By considering (1 - 2)P(2), show that all the zeroes
Let E be a field and , 6E E be nonzero polynomials. (a) If ab and a, prove that a = db for some nonzero d E E (b) If e gcd(a, b) and E Eis a common divisor of a and b of highest possible degree, prove that i= de for some nonzero d E E
i need the answer quickly
Consider the set S = {, 2n-1: m,n E Z} equipped with the usual addition and multiplication. (a) Does S contain an additive identity? (b) Does S contain a multiplicative identity? 1 (b) Show that is not in S. 4 (d) Is S a field under the above operations? Justify your answers.