Polynomial in the context of "Galois group"

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry.

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

The ring of integers of a number field K, denoted by O_K, is the intersection of K and A: it can also be characterized as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer if and only if the ring ${\displaystyle \mathbb {Z} [\alpha ]}$ is finitely generated as an abelian group, which is to say, as a ${\displaystyle \mathbb {Z} }$-module.

In mathematics, an algebraic equation or polynomial equation is an equation of the form ${\displaystyle P=0}$, where P is a polynomial, usually with rational numbers for coefficients.

For example, ${\displaystyle x^{5}-3x+1=0}$ is an algebraic equation with integer coefficients and

In mathematics, the term linear is used in two distinct senses for two different properties:

• linearity of a function (or mapping);
• linearity of a polynomial.

An example of a linear function is the function defined by ${\displaystyle f(x)=(ax,bx)}$ that maps the real line to a line in the Euclidean plane R that passes through the origin. An example of a linear polynomial in the variables ${\displaystyle X,}$ ${\displaystyle Y}$ and ${\displaystyle Z}$ is ${\displaystyle aX+bY+cZ+d.}$

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.

For example, the Diophantine equation ${\displaystyle 3x^{2}-2xy-y^{2}z-7=0}$ has an integer solution: ${\displaystyle x=1,\ y=2,\ z=-2}$. By contrast, the Diophantine equation ${\displaystyle x^{2}+y^{2}+1=0}$ has no such solution.

Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.

Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (${\displaystyle \pi }$) is a transcendental number.That is, ${\displaystyle \pi }$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if ${\displaystyle \pi }$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function ${\displaystyle f}$, is a member ${\displaystyle x}$ of the domain of ${\displaystyle f}$ such that ${\displaystyle f(x)}$ vanishes at ${\displaystyle x}$; that is, the function ${\displaystyle f}$ attains the value of 0 at ${\displaystyle x}$, or equivalently, ${\displaystyle x}$ is a solution to the equation ${\displaystyle f(x)=0}$. A "zero" of a function is thus an input value that produces an output of 0.

A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial ${\displaystyle f}$ of degree two, defined by ${\displaystyle f(x)=x^{2}-5x+6=(x-2)(x-3)}$ has the two roots (or zeros) that are 2 and 3.$${\displaystyle f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.}$$

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3, which is the number 9.In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x.The adjective which corresponds to squaring is quadratic.

The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) = x + 2x + 1.