Algebraic number in the context of "Geometry of numbers"

The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as ${\displaystyle {\sqrt {2}}}$ or ${\displaystyle 2^{1/2}}$. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction ⁠99/70⁠ (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.

In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (−), multiplication (×), division (÷), whole number powers, and roots (fractional powers).. For example, ⁠${\displaystyle 3x^{2}-2xy+c}$⁠ is an algebraic expression. Since taking the square root is the same as raising to the power ⁠1/2⁠, the following is also an algebraic expression:

$${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$$

In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. The quality of a number being transcendental is called transcendence.

Though only a few classes of transcendental numbers are known (partly because it can be extremely difficult to show that a given number is transcendental) transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers ⁠${\displaystyle \mathbb {R} }$⁠ and the set of complex numbers ⁠${\displaystyle \mathbb {C} }$⁠ are both uncountable sets, and therefore larger than any countable set.

In mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial ${\displaystyle g(x)\in K[x]}$ with coefficients in K such that g(a) = 0. Elements of A that are not algebraic over K are transcendental over K. A special case of an associative algebra over ${\displaystyle K}$ is an extension field ${\displaystyle L}$ of ${\displaystyle K}$.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of complex numbers and Q being the field of rational numbers).

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup ${\displaystyle \Gamma \subset G}$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G(A_F), for an algebraic group G and an algebraic number field F, is a complex-valued function on G(A_F) that is left invariant under G(F) and satisfies certain smoothness and growth conditions.

Klaus Friedrich Roth FRS (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.

Roth moved to England as a child in 1933 to escape the Nazis, and was educated at the University of Cambridge and University College London, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair at Imperial College London. He retired in 1988.