Abel and the Insolvability of the Quintic: Part 4

We now turn to the goal of this series namely to establish the fact that the general polynomial of degree $5$ or higher is not solvable by radicals over its field of coefficients. Here Abel's argument is quite terse and I have not been able to fully comprehend some parts of it. Also proof of some statements are not provided by Abel because it appeared quite obvious to him. We will provide here a proof which is based on Ruffini's arguments and its later simplification by Wantzel.

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Abel and the Insolvability of the Quintic: Part 3

The proof for the non-solvability of polynomial equation of degree $5$ (or more) by radicals obviously has to proceed via method of contradiction. Abel therefore assumed that such a solution was possible for a quintic and then figured out the most general form of such a solution. At the same time Abel observed that the radical expressions occurring in such a form must themselves be rational expressions of the roots desired. This was a key part which Abel proved for the first time. This result was later termed as the Theorem of Natural Irrationalities.

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Abel and the Insolvability of the Quintic: Part 2

In the last post we defined the concept of a radical field extension along the lines of the definition of algebraic functions given by Abel. In the current post we will study some properties of such field extensions which will ultimately enable us to study the field extension $\mathbb{C}(x_{1}, x_{2}, \ldots, x_{n})$ of $\mathbb{C}(s_{1}, s_{2}, \ldots, s_{n})$ where $s_{1}, s_{2}, \ldots, s_{n}$ are elementary symmetric functions of the indeterminates $x_{1}, x_{2}, \ldots, x_{n}$.

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Abel and the Insolvability of the Quintic: Part 1

Introduction

Most of the students come across the solution of linear and quadratic equations in their secondary classes. While the solution of a linear equation $ax + b = 0$ with $a, b$ being rational does not present any difficulties (because the solution $x$ itself turns out to be a rational number), a quadratic equation of the form $ax^{2} + bx + c = 0$ (with $a, b, c$ rational) does present significant challenges. For one thing the solution may not be rational and sometimes may not be even real. Usually one encounters the use of square roots to solve such an equation. Fortunately there is a standard formula for solving such equations $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ so that the equation can be solved directly in terms of its literal coefficients.

Many mathematicians tried to extend these ideas to solve the equations of third and fourth degrees. Thus during the 16th century Cardano solved the cubic and Ferrari solved the quartic equation. Later in the 18th century Lagrange published his classic work "Reflexions sur la resolution algébrique des equations" in which he unified the existing methods of solving equations upto degree $4$. He hoped that unifying all the available approaches into one coherent theory would help in solving higher degree equations. But neither Lagrange nor any other mathematician was able to provide a solution to quintics (equations of degree $5$) or higher degree equations. Then in 1824 a young Nowergian mathematician Niels Henrik Abel proved that it is not possible to solve a quintic equation in the same way as it is possible to solve equations of degree $2, 3$ or $4$.

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Fundamental Theorem of Algebra: Two Proofs

Introduction

Fundamental theorem of algebra is one of the most famous results provided in higher secondary courses of mathematics. Normally it is mentioned in chapter related to complex numbers where the reader is made aware of the power of complex numbers in solving polynomial equations. The theorem guarantees that any non-constant polynomial with real or complex coefficients has a complex root.

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Matrix Inversion: Partition Method

Introduction

Today we will discuss a not-so-famous method of inverting matrices. This method is recursive in the sense that given a method to find inverse of square matrix of order $ n$ it can be applied to find the inverse of a matrix of order $ (n + 1)$. This method is named Partition Method or the Escalator Method. The idea is to partition a matrix into smaller sub-matrices and then calculate the inverse from the given inverse of one of the smaller sub-matrices.

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Logarithms using Square Roots

Introduction

In the last post we discussed that a proper theory of logarithmic function can not be developed without using the analytical approach (or methods of calculus in simpler language). Here we discuss more about the common logarithms (i.e. logarithms to the base 10) which are normally introduced to students in secondary classes without using any calculus. The basic idea is simple enough: if $ y = 10^{x}$ then we write $ x = \log_{10}y$ and say that $ x$ is the common logarithm of $ y$. The difficulty with this approach is that the meaning of $ 10^{x}$ cannot be explained properly without using calculus if $ x$ is irrational. However in the examples and exercises given the exponent is either a symbol (like $ a, b, c, x$) or is a rational number so there is no confusion when the concept of logarithm is presented via this approach. And the student is able to learn the fundamental properties of logarithms and their practical usage in computation with relative ease.

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The Mysterious Rank (of a Matrix) Demystified

Let us start with the following system of linear equations (written in matrix form):
$$ AX = B\text{ where } A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\, X = \begin{bmatrix} x_{1} \\ x_{2} \\ \cdots \\ x_{n} \end{bmatrix},\, B = \begin{bmatrix} b_{1} \\ b_{2} \\ \cdots \\ b_{m} \end{bmatrix}$$ The augmented matrix of the system is: $$\tilde{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & b_{1} \\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{bmatrix}$$

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The Mysterious Rank (of a Matrix): Elementary Row Operations

In the previous post we formed notions of matrix, determinants, rank of a matrix and system of linear equations. We also mentioned the fundamental theorem on solution of system of linear equations which tells us the importance of rank in deciding the nature of solution of such systems of equations. It is now time to establish the theorem and to do that we start by defining a systematic method of solving such systems of linear equations.

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The Mysterious Rank (of a Matrix)

In this post I will be talking about the simple and beautiful concept of matrices. In particular I will discuss about the rank of a matrix and its usage in solving system of linear equations. In order to keep the post to a reasonable length I will not dwell upon the usual operations on matrices and assume that the reader is familiar with them, but a paragraph or two about these operations definitely makes sense.

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Field Automorphisms: A Nice Touch on "Ambiguity"

I am sure that the title of the post is going to confuse many readers, but for lack of a better title please be content with it. Rest assured that the post itself is quite unambiguous. We would like to throw some light on the a special kind of ambiguity which we find in mathematical systems (here mainly in algebra).

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A Taste of Modern Algebra: Remainder Theorem for Polynomials

Introduction

In high school curriculum we are taught the "Remainder Theorem" as one of the important results of Algebra. The theorem and its proof are quite simple, but the practical application and theoretical ramifications of this result are quite interesting. To recall, the theorem is stated as follows:

Remainder Theorem: If $ f(x) = a_{0}x^{n} + a_{1}x^{n - 1} + \cdots + a_{n - 1}x + a_{n}$ is a polynomial with real coefficients then the remainder obtained on dividing $ f(x)$ by $ (x - a)$ is $ f(a)$.

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Gauss and Regular Polygons: Conclusion

The Central Result

This is the concluding post in this series and we aim to prove the following result (proved in part by Gauss and finally the converse by Wantzel):
A regular polygon of $ n, n > 2$ sides can be constructed by Euclidean tools if and only if $ \phi(n) = 2^{k}$.

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Gauss and Regular Polygons: Gaussian Periods Contd.

Properties of Gaussian Periods

In this post we are going to establish the following properties of the Gaussian periods which will ultimately lead to a solution of the equation $ z^{p} - 1 = 0$. Again as in previous post, $ p$ is to be considered a prime unless otherwise stated. In the following we have $ e, f$ as two positive integers with $ ef = (p - 1)$.

1. Any period of $ f$ terms can be expressed as a polynomial in any other period of $ f$ terms with rational coefficients.
2. If $ g$ divides $ (p - 1)$ and $ f$ divides $ g$, then any period of $ f$ terms is a root of a polynomial equation of degree $ g / f$ whose coefficients are rational expressions of a period of $ g$ terms.

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Gauss and Regular Polygons: Gaussian Periods

Introduction

In order to solve the equation $ z^{n} - 1 = 0$ Gauss introduced some sums of the $ n^{th}$ roots of unity which he called periods, and using these periods he was able to reduce the solution of $ z^{n} - 1 = 0$ to a sequence of solutions of equations of lower degrees. The technique offered by Gauss is extremely beautiful and completely novel and it uses the symmetry between the various $ n^{th}$ roots of unity to achieve the final solution.

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Gauss and Regular Polygons: Cyclotomic Polynomials

Introduction

The word "Cyclotomy" literally means "cutting a circle". So the subtitle of the post suggests that the post is going to be about some polynomials which are related to cutting a circle. Cutting a circle actually refers to dividing a given circle into a number of arcs of same length. Supposing that we are able to divide a given circle into, say $ n$, arcs of equal length by means of points $ P_{0}, P_{1}, \ldots, P_{n - 1}$ then joining the adjacent points we obtain a regular polygon $ P_{0}P_{1}\ldots P_{n - 1}$ of $ n$ sides. Therefore cyclotomic polynomials are somehow related to the construction of regular polygons.

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Gauss and Regular Polygons: Complex Numbers

Introduction to Complex Numbers

Complex numbers are not really complex! In fact they are reasonably simple to understand and operate upon. The concept is definitely strange on a first look, but is damn powerful and has diverse ramifications in various branches of mathematics. Now, to illustrate the point that these numbers are really simple, we are gonna define them in terms of quantities already known.

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Gauss and Regular Polygons: Euclidean Constructions Primer

Introduction

What do we exactly mean by the term "Euclidean Constructions"? Informally the term refers to the geometrical constructions done using the ruler (also called straightedge) and compass. Such constructions are studied as part of high-school (7th to 10th grade) mathematics curriculum and I hope most readers are familiar with the construction of bisection of line segment, bisection of an angle and construction of equilateral triangles.

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Gauss and Regular Polygons

Introduction

After studying elliptic integrals and formulas for $\pi$, we shall now focus on one of the most beautiful gems discovered by Gauss at the age of 17. Gauss proved that the construction of a regular polygon of 17 sides is possible by using an unmarked ruler and a compass only (henceforth these will be known as Euclidean tools and such constructions will be called Euclidean constructions). This is quite remarkable because since 2000 years or so from the time of Euclid the only polygons which were constructible in such a fashion were having sides 3, 4, 5, 6, 8, 10, 12, 15. Gauss added a new number in this series namely 17 and generalized his results to add further numbers. Legend has it that Gauss was so excited by this discovery that he decided to make a career as a mathematician.

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