Galois Theory

Galois theory is a branch of algebra that studies the relationship between field extensions and groups. It was developed by the French mathematician Évariste Galois in the 19th century, and has since become a fundamental tool in many areas of mathematics and physics.

One of the main goals of Galois theory is to understand the symmetries of polynomial equations. For example, consider the polynomial equation $x^2 + 1 = 0$. This equation has no solutions in the real numbers, but it does have solutions in the complex numbers. In fact, the solutions are $\pm i$, the imaginary unit. We can think of these solutions as the roots of the polynomial, and we can write the equation as $(x - i)(x + i) = 0$.

Now consider the polynomial equation $x^3 - 2 = 0$. This equation also has no solutions in the real numbers, but it does have a solution in the complex numbers. We can write the solution as $x = \sqrt[3]{2} \cdot \omega$, where $\omega$ is a complex number satisfying $\omega^3 = 1$. In fact, there are two other complex numbers that satisfy this equation as well, namely $\sqrt[3]{2} \cdot \omega^2$ and $\sqrt[3]{2}$. These three solutions correspond to the roots of the polynomial, and we can write the equation as $(x - \sqrt[3]{2} \cdot \omega)(x - \sqrt[3]{2} \cdot \omega^2)(x - \sqrt[3]{2}) = 0$.

Notice that in both of these examples, the solutions are related to each other by symmetries. For example, in the first equation, the solutions are symmetric with respect to the imaginary axis. In the second equation, the solutions are related by the symmetries of an equilateral triangle in the complex plane. Galois theory is concerned with understanding these symmetries and how they relate to the structure of the polynomial equations and their solutions.

The key idea of Galois theory is to associate to each polynomial equation $f(x)$ a group of symmetries, called the Galois group, which reflects the symmetries of the roots of the polynomial. The Galois group encodes information about the field extensions generated by the roots of the polynomial, and it provides a way to understand the properties of these extensions.

For example, consider the polynomial equation $x^2 - 2 = 0$. We know that the solutions to this equation are $\pm \sqrt{2}$. The Galois group of this equation consists of two elements: the identity element and the automorphism of the field $\mathbb{Q}(\sqrt{2})$ that maps $\sqrt{2}$ to $-\sqrt{2}$. This Galois group reflects the fact that there are exactly two field extensions of $\mathbb{Q}$ that contain the roots of the polynomial: $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(-\sqrt{2})$.

Galois theory also provides a criterion for determining whether a polynomial equation is solvable by radicals, that is, whether its solutions can be expressed in terms of elementary functions and field operations. This criterion is based on the Galois group of the equation, and it states that a polynomial equation is solvable by radicals if and only if its Galois group is a solvable group.

In summary, Galois theory provides a powerful tool for understanding the symmetries of polynomial equations and their solutions. It has applications in many areas of mathematics and physics, including number theory, algebraic geometry, and quantum mechanics.

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