Question: Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b

Let F/K be a field extension and a, b E F be

Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has degree n over K, prove that the elements a+b, ab, ab, ab-1 have degree at most mn over K.

Step by Step Solution

★★★★★

3.53 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

To prove that the elements a b ab a b and ab 1 have degree at most mn over K we need to show that their minimal polynomials have degree at most mn Let ... View full answer

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!

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)...

Proof that Let K be a field such that for every extension field F the maximal algebraic extension of K contained in F is K itself. Then K is algebraically closed.

Part (ii) or (ii)' of the Fundamental Theorem (2.5) is equivalent to: an intermediate field E is normal over K if and only if the corresponding subgroup E' is normal in G = Aut K F in which case G i...

Instructions Demographics: Part 1: Demographics Info and You Video Watch the Demographics Info and You Video and take notes. 1. Give a page overview of the major changes predicted for the US with...

Let K/k be a cubic extension of fields of characteristic 0, of the form K = k(sqrt3(d)) for some d belongs to k that is not a cube in k. We assume ...(see the attachment) Let K / k: be a cubic...

1. Let V be the real vector space of 3 x 3 symmetric matrices. A typical element of V looks like a d d b e f for some a, b, c, d, e, f E R. Let W be the real vector space of degree 3 polynomials in...

Please consider the data for SKUs X, Y, and Z. The store currently carries SKUs X and Y. The store manager considers discontinuing SKUs X and Y in favor of a new SKU Z, expecting that all consumers...

What is the purpose of the QUADRATIC FORMULA? A Determine the distance between two points B Calculate the axis of symmetry of a quadratic function Determine the zeros of a quadratic function D...

ls solution Yes or No ? 3 The firsite field Fas a splitting field of the polyriorrial x (4-1) -1. 4 The Galois group Gal(Q2/5.j): Q) is of order 6 and it is isomorphic to the permutation group S3...

The Basics of Financial Mathematics Spring 2003 Richard F. Bass Department of Mathematics University of Connecticut c These notes are 2003 by Richard Bass. They may be used for personal use or class...

Portal Corporation produces the same power generator in two Illinois plants, a new plant in Peoria and an older plant in Moline. The following data are available for the two plants. All fixed costs...

a) A software company has hired new staffs with particular expertise as shown on the Table (1): Table (1) Expertise Number of Staff Python Programming 20 R Programming 15 Java Programming 10 i. In...

Which of the following firms pursue a Broad strategy? ( Please identify all that apply ) . Option A Miller Option B Stroh s Option C Coors Option D Pabst Option E Anheuser - Busch Option F Heileman

Repeat Problem 23 using MATLAB. Data From Problem 23: For the unity feedback system of Figure P6.3 with determine the range of K for stability. K(s + 3)(s + 5) G(s) : (s- 2)(s- 4) %3D

Yosemite Corp. has an outstanding debt of $10 million on which it pays a 5 percent fixed interest rate annually. Yosemite just made its annual interest payment and has three years remaining until...

What is the minimum number of test cases to ensure that every statement in Example 6.3 is tested? Data from Example 6.3 Source Code for the Sample Function That Was Specified in Example 6.2 i = 1; //...

What is the problem with quotes in an SQL command obtained from a form element in an HTML document?

Derive a demand curve using indifference curve analysis.

Suppose you need to know an equation of the tangent plane toa surface S at the point Ps2,1,3d. You dont have an equation

(a) If find [F: Q] and a basis of F over Q. (b) Do the same for where i C, i 2 = -1, and is a complex (nonreal) cube root of 1. F = Q(,),

(a) If A and Bare R-modules, then the set Hom R (A,B) of all R-module homomorphisms A B is an abelian group with + g given on a A by(+ g)(a) = f(a) + g(a) B. The identity element is the zero map....

Proof that If a group G is the (internal) direct product of its subgroups H, K, then H G/ K and G/H K.

What are the main benefits of JIT/lean? Name a company using JIT/lean, and describe how JIT/lean contributes to the company *s competitive advantage.

Is tire service system of WestJet JIT/lean?

What is value stream mapping and why is it important?