a complex number z is said to be algebraic if its root of a polynomial that...

a complex number z is said to be algebraic if its root of a polynomial that has inteher coefficients. Let A be the collection of algebraic numbers. Show that A is countable.

Expert Solution

Colby Messinger answered 2 years ago

A polynomial in Z[x] is said to be primitive if the greatest common divisor of its...

A polynomial in Z[x] is said to be primitive if the greatest common divisor of its coefficients is 1. Prove the product of two primitive polynomials is primitive. [Hint: Use proof by contradiction.]

If p(z) is a polynomial of degree n and that if α is a root of...

If p(z) is a polynomial of degree n and that if α is a root of p(z) = 0, then p(z) factors as p(z) = (z−α)q(z) where q(z) has degree (n − 1). Use this and induction to show that a polynomial of degree n has at most n roots.

Using Newton-Raphson method, find the complex root of the function f(z) = z 2 + z...

Using Newton-Raphson method, find the complex root of the function f(z) = z 2 + z + 1 with with an accuracy of 10–6. Let z0 = 1 − i. write program c++ or matlab

Complex Numbers: What's the difference between Log(z), log(z) and ln(z)? where z is a complex number,...

Complex Numbers: What's the difference between Log(z), log(z) and ln(z)? where z is a complex number, z = x + iy

In Matlab: Any complex number z=a+bi can be given by its polar coordinates r and θ,...

In Matlab: Any complex number z=a+bi can be given by its polar coordinates r and θ, where r=|z|=sqrt(a^2+b^2) is the magnitude and θ= arctan(ba) is the angle. Write a function that will return both the magnitude r and the angle θ of a given complex numberz=a+bi. You should not use the built-in functions abs and angle. You may use the built-in functions real and imag.

Roots of a complex number Find all the solutions to the equation (a) z^ 4 −...

Roots of a complex number Find all the solutions to the equation (a) z^ 4 − 1 = 0 (b) z ^5 + 2^5 i = 0 (c) z^ 6 − 16z^ 3 + 128 = 0

Show that any polynomial over C (the complex numbers) is the characteristic polynomial of some matrix...

Show that any polynomial over C (the complex numbers) is the characteristic polynomial of some matrix with complex entries. Please use detail and note any theorems utilized.

Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

Does every polynomial equation have at least one real root? a. Why must every polynomial equation...

Does every polynomial equation have at least one real root? a. Why must every polynomial equation of degree 3 have at least one real root? b. Provide an example of a polynomial of degree 3 with three real roots. How did you find this? c. Provide an example of a polynomial of degree 3 with only one real root. How did you find this?

in hyperbolic/modern geometry. Let C be a circle and z any complex number. Prove that the...

in hyperbolic/modern geometry. Let C be a circle and z any complex number. Prove that the point z* symmetric to z with respect to C is unique.

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