Generator For A Nasty Number Field

Introduction

In the realm of algebraic numbers, finding a generator for a number field can be a daunting task, especially when dealing with complex and nasty number fields. A generator, in this context, is an element in the number field that can be used to express all other elements in the field. In this article, we will delve into the world of algebraic numbers and explore the concept of generators for a nasty number field.

What are Algebraic Numbers?

Algebraic numbers are complex numbers that are roots of non-constant polynomials with rational coefficients. In other words, they are numbers that can be expressed as the solution to a polynomial equation with rational coefficients. For example, the square root of 2 is an algebraic number because it is the root of the polynomial equation x^2 - 2 = 0.

Minimal Polynomials

A minimal polynomial of an algebraic number is the simplest polynomial with rational coefficients that has the number as a root. In other words, it is the polynomial with the smallest degree that has the number as a root. For example, the minimal polynomial of the square root of 2 is x^2 - 2.

A Collection of Algebraic Numbers

We have a collection of algebraic numbers, which we will list along with their minimal polynomials:

Algebraic Number	Minimal Polynomial
$a_1 = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$	$p_1(x) = 2x^4 - 4x^3 + 4x^2 - 2x + 1$
$a_2 = \sqrt[3]{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$	$p_2(x) = x^3 - \frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)$
$a_3 = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)} + \sqrt[3]{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$	$p_3(x) = 2x^4 - 4x^3 + 4x^2 - 2x + 1$

Finding a Generator

To find a generator for a nasty number field, we need to find an element in the field that can be used to express all other elements in the field. In other words, we need to find an element that can be used to generate all other elements in the field.

Theoretical Background

Theoretical background for finding a generator for a nasty number field involves the use of Galois theory and the concept of minimal polynomials. Galois theory provides a way to study the symmetries of a field and the minimal polynomials provide a way to express the elements of the field in terms of a single generator.

Algorithm for Finding a Generator

The algorithm for finding a generator for a nasty number field involves the following steps:

1. Find the minimal polynomial: Find the minimal polynomial of the algebraic number.
2. Find the roots of the minimal polynomial: Find the roots of the minimal polynomial.
3. Check if the roots are in the field: Check if the roots are in the field.
4. Find a generator: Find a generator that can be used to express all other elements in the field.

Implementation

The implementation of the algorithm for finding a generator for a nasty number field involves the use of computer algebra systems such as Mathematica or Sage.

Example

Let's consider the algebraic number $a_1 = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$ with minimal polynomial $p_1(x) = 2x^4 - 4x^3 + 4x^2 - 2x + 1$. We can use the algorithm for finding a generator to find a generator for this number field.

Step 1: Find the minimal polynomial

The minimal polynomial of $a_1$ is $p_1(x) = 2x^4 - 4x^3 + 4x^2 - 2x + 1$.

Step 2: Find the roots of the minimal polynomial

The roots of $p_1(x)$ are $x = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$, $x = -\sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$, $x = \sqrt[4]{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$, and $x = -\sqrt[4]{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$.

Step 3: Check if the roots are in the field

The roots are in the field.

Step 4: Find a generator

A generator for this number field is $a_1 = \sqrt{\frac{1}{2} \left(1+2 \sqrt{2}+\sqrt{3}+\sqrt{6}\right)}$.

Conclusion

Introduction

In our previous article, we discussed the concept of generators for a nasty number field and provided an algorithm for finding a generator. In this article, we will answer some frequently asked questions about generators for a nasty number field.

Q: What is a generator for a nasty number field?

A: A generator for a nasty number field is an element in the field that can be used to express all other elements in the field. In other words, it is an element that can be used to generate all other elements in the field.

Q: How do I find a generator for a nasty number field?

A: To find a generator for a nasty number field, you need to follow the algorithm we provided in our previous article. This involves finding the minimal polynomial, finding the roots of the minimal polynomial, checking if the roots are in the field, and finding a generator.

Q: What is the minimal polynomial?

A: The minimal polynomial is the simplest polynomial with rational coefficients that has the number as a root. In other words, it is the polynomial with the smallest degree that has the number as a root.

Q: How do I find the minimal polynomial?

A: To find the minimal polynomial, you can use a computer algebra system such as Mathematica or Sage. You can also use the Rational Root Theorem to find the possible roots of the polynomial and then use synthetic division to find the minimal polynomial.

Q: What are the roots of the minimal polynomial?

A: The roots of the minimal polynomial are the values that satisfy the polynomial equation. In other words, they are the values that make the polynomial equal to zero.

Q: How do I check if the roots are in the field?

A: To check if the roots are in the field, you need to verify that the roots are algebraic numbers. You can do this by checking if the roots are roots of a non-constant polynomial with rational coefficients.

Q: How do I find a generator?

A: To find a generator, you need to find an element in the field that can be used to express all other elements in the field. This can be done by using the roots of the minimal polynomial and the properties of the field.

Q: What are some common mistakes to avoid when finding a generator?

A: Some common mistakes to avoid when finding a generator include:

• Not finding the minimal polynomial correctly
• Not checking if the roots are in the field
• Not using the correct properties of the field to find a generator

Q: Can I use a computer algebra system to find a generator?

A: Yes, you can use a computer algebra system such as Mathematica or Sage to find a generator. These systems can perform the calculations and provide the results.

Q: How long does it take to find a generator?

A: The time it takes to find a generator depends on the complexity of the field and the number of elements in the field. In general, it can take anywhere from a few minutes to several hours or even days to find a generator.

Conclusion

In conclusion, finding a generator for a nasty number field involves the use of Galois theory and the concept of minimal polynomials. The algorithm for finding a generator involves finding the minimal polynomial, finding the roots of the minimal polynomial, checking if the roots are in the field, and finding a generator. By following the algorithm and avoiding common mistakes, you can find a generator for a nasty number field.