Question: Let f(x) = F[x] be a nonconstant polynomial and a be an element in any field extension E of F. Then, a is a multiple

Let f(x) = F[x] be a nonconstant polynomial and a be

Let f(x) = F[x] be a nonconstant polynomial and a be an element in any field extension E of F. Then, a is a multiple root of f(x) if and only if f(a) = 0 = f'(a); that is, a is a root of both f'(x) and f'(x).

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Give a two- or three-sentence synopsis of Theorem 29.3. Data from Theorem 29.3 Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there exists an extension field E of F and an ...

1. (60 pts True/False. Indicate whether the statement is true or false. If the statement is false, present an explanation or counterexample. (No explanation necessary for each true statement.) (2...

,, kindly solve Consider the Linear Regression Model 1. For any X = x, let Y = x8 + U, where B E R*. 2. X is exogenous. 3. The probability model is { f (u; ) is a distribution on R : E, [U] = 0, Var,...

Hello, can someone answer these two questions? I'm stuck. THANKS! Exercise 3.3. Consider the field F = Z5. Let f = X 3 +X+ [1] F[X], g = [2]X 2 + [3]X+ [4] F[X], and h = [3]X 2 + [2]X + [1] F[X]....

s1 educated (SSE) student for every three public school educated (PSE) students. Reasoning that students are not very dissimilar from threads, he suggests the following entry and exit routines be...

Let F be an algebraic closure of the field Q of rational numbers and let E F be a splitting field over Q of the set S = { x 2 + a | a Q} so that E is algebraic and Galois over Q (Theorem 3.11). (a)...

Mark each of the following true or false. ____ a. The number is transcendental over Q. ____ b. C is a simple extension of R ____ c. Every element of a field F is algebraic over F. ____ d. R is an...

Theeren ; "-x = TTI, (x) where fi (x)E (F. [ x] are the manic irreducible polynomials with dey (5, ) | n Proof Suppose flx) ElFLAJ is manic & irreducible & dog (f) In Then K' FPL x] / (food) is a...

How we proceed f(1). Find a polynomial g(x) in Z[x] such that Ker d = (g(x)). Is there more than one possibility for g(x)? To what familiar ring is Z[x]/Ker o isomorphic? Do this exercise with Z...

Give Correct ANSWERS Human-Computer Interaction (a) If you had been one of the original inventors of the WIMP interface, and engineers on the technical team had been sceptical about the advantages...

The Statehas a legally separate State Building Authority which has a board appointed bythe Governor. The Authority issues debtin its own name, holds title to buildings in its own name, and leases...

Grand Canyon Tours Co. is a travel agency. The nine transactions recorded by Grand Canyon Tours during April 2014, its first month of operations, are indicated in the following T accounts: Indicate...

The "simplest" way to alter the separation in GC is to alter the temperature program in the oven. The diagram bellow represents: Gradient temperature program Isothermal temperature program

A hula hoop and solid disc of equal radii are rolled, from rest and without slipping, down an incline plane. Which one will reach the bottom of the inclined plane first? Explain.

Where would you be more comfortable? An organization with a performance review approach more like the Boston Consulting Group or more like Netflix? Why?

What are some work activities in which the acceptable range of variation might be higher than average? What about lower than average? (Hint: Think in terms of the output from the work activities,...

What type (or types) of controlfeedforward, concurrent, or feedbackdo you think would have been helpful in this situation? Explain your choice.