What is mathematics?

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What is mathematics?

Mathematics is the science and study of quantity ,structure ,space and change. There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics “the science that draws necessary conclusions”. Albert Einstein, on the other hand, stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” The use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Today, mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences.

WHAT IS AN ALGEBARIC INTEGER?

In number theory, an algebraic integer is a complex number that is a root of some monic polynomial (leading coefficient 1) with coefficients in Z. The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. If c is a rootof monic .integral polynomial of degree d, namely a root of a polynomial of the form

F(x)=x+………

And is the root of a polynomial of degree less than d, then is called an algebraic integer of degree d.

WHAT IS ALGEBRAIC NUMBER THEORY?

In mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures. Algebraic number theory involves using techniques from algebra and finite group theory to gain a deeper understanding of number fields. The main objects that we study in algebraic number theory are number fields, rings of integers of number fields, ideal class groups, prime ideals.

APPLICATIONS OF ALGEBARIC NUMBER THEORY

Pell’s Equation=>Units in real quadratic fields =>Unit

groups in number fields

Pell’s equationis anyDiophantine equationof the form

wherenis anon squareinteger andxandyare integers. Trivially,x= 1 andy= 0 always solve this equation.Lagrangeproved that for anynatural numbernthat is not aperfect squarethere arexandy> 0 that satisfy Pell’s equation. Moreover, infinitely many such solutions of this equation exist. These solutions yield goodrationalapproximations of the formx/y to thesquare rootofn.

* Diophantine Equations => For which n does have a non trivial solution?

Inmathematics,aDIOPHANTINE EQUATION anindeterminate polynomialequationthat allows the variables to beintegersonly. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define analgebraic curve,algebraic surfaceor more general object.

_

* Integer Factorization =>Factorization of ideals

Innumber theory,integer factorizationorprime factorizationis the breaking down of acomposite numberinto smaller non-trivialdivisors, which when multiplied together equal to the original integer.

When the numbers are very large, no efficient integerfactorizationalgorithmis publicly known; then we use the algebraic number theory results.

Riemann Hypothesis => Generalized Riemann Hypothesis.

The Riemann hypothesis implies results about the distribution of prime numbers that

are in some ways as good as possible. Along with suitable generalizations, it is considered by many mathematicians to be the most important unresolved problem inpure mathematics

* Wiles’s proof of Fermat’s Last Theorem, i.e., has no nontrivial

integer solutions, uses methods from algebraic number theory extensively (in

addition to many other deep techniques). Attempts to prove Fermat’s Last

Theorem long ago were hugely influential in the development of algebraic

number theory (by Dedekind, Kummer, Kronecker).In number theory,Fermat’s Last Theoremstates that no threepositiveintegersa,b, andccan satisfy the equation an+bn=cnfor any integer value ofngreater than two. This theorem was firstconjecturedbyPierre de Fermatin 1637, but was not proven until 1995 despite the efforts of many mathematicians. The unsolved problem stimulated the development ofalgebraic number theoryin the 19th century and the proof of themodularity theoremin the 20th. It is among the most famous theorems in thehistory of mathematics.

* Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields.

SOME OTHER EXAMPLES OF ALGEBABIC NUMBER THEORY

An element a of a field K is said to be a primitive nth root of 1 if but for any d < n, i.e., if a is an element of order n in K. The nth roots of 1 in C are the numbers , 0,and is a primitive nth root of 1 if and only if m is relatively prime to n.

Let a be a primitive nth root of 1. Then am is again a primitive nth root of 1 if and only if m is relatively prime to n.

F be a field. F(x) f[x] is irreducible iff f(x+1) is irrehucible.

WHAT IS FIELD AIRTHMATIC?

In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite .

Algebaric number theory and algebaric geometry has been already discussed.

Model theory:-Inmathematics,model theoryis the study of (classes of) mathematicalstructuressuch asgroups,fields,graphs, or even universes ofset theory, using tools frommathematical logic. A structure that gives meaning to the sentences of a formal language is called amodelforthe language. If a model for a language moreover satisfies a particular sentence or theory (set of sentences), it is called a modelofthe sentence or theory. Model theory has close ties toalgebraanduniversal algebra.

WHAT IS FINITE FIELD?

Finite fieldis different from standard integerarithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field. While each finite field is itself not infinite, there are infinitely many different finite fields; their number of elements is necessarily of the form pn wherepis a prime numberandnis apositive integer, and two finite fields of the same size are isomorphic. The primepis called thecharacteristicof the field, and the positive integernis called the dimensionof the field over itsprime field.

Introduction to finite fields

Fields are abstractions of familiar number systems (such as the rational numbers Q, the real numbers R, and the complex numbers C) and their essential properties. They consist of a set F together with two operations, addition (denoted by +) and multiplication (denoted by ), that satisfy the usual arithmetic properties:

  1. (F,+) is an abelian group with (additive) identity denoted by 0.
  2. (F{0}, ) is an abelian group with (multiplicative) identity denoted by 1.
  3. The distributive law holds: (a+b) c = a c+b c for all a,b, c ∈ F.

If the set F is finite, then the field is said to be finite.

This section presents basic facts about finite fields. Other properties will be presented throughout the book as needed.

Field operations

A field F is equipped with two operations, addition and multiplication. Subtraction of field elements is defined in terms of addition: for a,b ∈ F, a −b = a +(−b) where −b is the unique element in F such that b+(−b) = 0 (−b is called the negative of b). Similarly, division of field elements is defined in terms of multiplication: for a,b ∈ F

With b, where is the unique element in F such that b = 1.

( is called the inverse of b.)

Existence and uniqueness

The order of a finite field is the number of elements in the field. There exists a finite field F of order q if and only if q is a prime power, i.e., q = pm where p is a prime number called the characteristic of F, and m is a positive integer. If m = 1, then F is called a prime field. If m ≥ 2, then F is called an extension field. For any prime power q, there is essentially only one finite field of order q; informally, this means that any two finite fields of order q are structurally the same except that the labeling used to represent the field elements may be different . We say that any two finite fields of order q are isomorphic and denote such a field by Fq .

Prime fields

Let p be a prime number. The integers modulo p, consisting of the integers

{0,1,2, . . ., p −1} with addition and multiplication performed modulo p, is a finite field of order p.We shall denote this field by Fp and call p the modulus of Fp. For any integer a, a mod p shall denote the unique integer remainder r, 0 ≤r ≤ p−1, obtained upon dividing a by p; this operation is called reduction modulo p.

Example (prime field F29) The elements of F29 are {0,1,2, . . .,28}. The following

are some examples of arithmetic operations in F29.

  1. Addition: 17+20 = 8 since 37 mod 29 = 8.
  2. Subtraction: 17−20 = 26 since −3 mod 29 = 26.
  3. Multiplication: 17 20 = 21 since 340 mod 29 = 21.
  4. Inversion: 17−1 = 12 since 17 12 mod 29 = 1.

Binary fields

Finite fields of order 2m are called binary fields or characteristic-two finite fields. One way to construct is to use a polynomial basis representation. Here, the elements of are the binary polynomials (polynomials whose coefficients are in the field

F2 = {0,1}) of degree at most m −1:

= { : {0,1}}.

An irreducible binary polynomial f (z) of degree m is chosen

Irreducibility of f (z) means that f (z) cannot be factored as a product of binary polynomials each of degree less than m. Addition of field elements is the usual addition of polynomials, with coefficient arithmetic performed modulo 2. Multiplication of field elements is performed modulo the reduction polynomial f (z). For any binary polynomial a(z), a(z) mod f (z) shall denote the unique remainder polynomial r (z) of degree less than m obtained upon long division of a(z) by f (z); this operation is called reduction modulo f (z).

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