Q:Information: - Evariste Galois (25 October 1811 31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel. - Hasse 's theorem on elliptic curves , also referred to as the Hasse bound , provides an estimate of the number of points on an elliptic curve over a finite field , bounding the value both above and below . If N is the number of points on the elliptic curve E over a finite field with q elements , then Helmut Hasse 's result states that N - ( q +1 ) \ le 2 \ sqrt ( q ) . That is , the interpretation is that N differs from q + 1 , the number of points of the projective line over the same field , by an ' error term ' that is the sum of two complex numbers , each of absolute value q. This result had originally been conjectured by Emil Artin in his thesis . It was proven by Hasse in 1933 , with the proof published in a series of papers in 1936 . Hasse 's theorem is equivalent to the determination of the absolute value of the roots of the local zeta - function of E. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve . - In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - In mathematics, a finite field or Galois field (so-named in honor of Evariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. - An integer (from the Latin "integer" meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and2048 are integers, while 9.75, , and are not. - A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 3, 1 1 3, etc. are all valid factorizations of 3. - Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. - In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a higher dimensional space or defined over some more general field. After reading the paragraphs above, we are interested in knowing the entity with which 'hasse's theorem on elliptic curves' exhibits the relationship of 'subclass of'. Find the answer from the choices below. Choices: - algebra - algebraic curve - analysis - arabic - definition - economics - engineering - equation solving - factorization - french - greek - integer - latin - learning - mathematician - mathematics - medicine - mod - natural - natural number - number - plane - polynomial - problem - professional - quantity - range - set - study - theorem - theory - work
A:
algebraic curve