19.2.8. Let F7 = Z/7Z be the field with seven elements, and let f,g Є F7[x] bedefined by f(x) = x³ + x³ + x + 1 and g(x) = x² + x. Is f in the idealgenerated by g? If the answer is no, then find h = F7[×] with degree lessthan two such that f − h is in the ideal generated by g.

Step 1:Theorem- Suppose 𝐹 is a field, and 𝑓(𝑥)∈𝐹[𝑥], and `I=𝑖𝑠𝑖𝑑𝑒𝑎𝑙 ≥ ≠𝑟𝑎𝑡𝑒𝑑𝑏𝑦 f.…

Answered: 19.2.8. Let F7 = Z/7Z be the field with…

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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1. Let F := 2/2z in each item below an element a and a polynomial f(X) € F[X] satisfyingin f(a) = 0 is given. Write down a multiplication table for the elements of the finite field K = F(a) i. a is a zero of f(X) = X²+X+1 ii. a is a zero of f(X) = X³ + X + 1 iii. a is a zero of f(X) = X² + 1
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].
Let R = F_2[X], the ring of polynomials with coefficients from the field with two elements. Let J be the ideal generated by X^3+1, so specifically it is the set of all polynomials divisible by X^3+1. What is the product (X^2+J).(X^2+J) in the factor ring R/J? a. X b. 0+J C. 0 d. X+J e. Not sure
5.9. (a) Let K/F be an extension of fields. Prove that [K: F] = 1 if and only if K = F. (b) Let L/F be a finite extension of fields, and suppose that [L: F] is prime. Suppose further that K is a field lying between F and L; i.e., F C K C L. Prove that either K = F or K = L.
Show that the cubic field generated by a root of f(x) = %3D x3 -3x2 - 10x - 8, where f is irreducible over Q, has no power base. Use the fact that disc(f)=(-503)(4).
In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer}. 1) Prove that I is an ideal of Z[x]. 2) What is the elements of Z[x]/1?
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.
1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
(b) Prove that Z/4Z is not a field.
Need a and b

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19.2.8. Let F7 = Z/7Z be the field with seven elements, and let f,g Є F7[x] be defined by f(x) = x³ + x³ + x + 1 and g(x) = x² + x. Is f in the ideal generated by g? If the answer is no, then find h = F7[×] with degree less than two such that f − h is in the ideal generated by g.