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In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields.
A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size.
An Arakelov divisor (or replete divisor [4]) on a global field is an extension of the concept of divisor or fractional ideal. It is a formal linear combination of places of the field with finite places having integer coefficients and the infinite places having real coefficients. [3] [5] [6] Arakelov height
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a 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 groups .
If the valuation is Archimedean, one obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension /: a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems.
For example, the infinite sequence (,, … ) {\displaystyle (1,2,\ldots )} of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + ∞ {\displaystyle +\infty } as its least upper bound and as its limit (an actual infinity).
More generally, for any square-free integer , the quadratic field is a number field obtained by adjoining the square root of to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, d = − 1 {\displaystyle d=-1} .