Prime number

In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. Or for short: A prime number is a natural number with exactly two natural divisors. A natural number that is greater than one and is not a prime is called a composite number. The numbers zero and one are neither prime nor composite. The property of being a prime is called primality. Prime numbers are of fundamental importance in number theory.

Generalizations

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.

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Prime elements in rings

One can define prime elements and irreducible elements in any integral domain. For the ring Z of integers, the set of prime elements equals the set of irreducible elements; it's {...−11, −7, −5, −3, −2, 2, 3, 5, 7, 11, ...}.

Related Topics:
Prime element - Irreducible element - Integral domain

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As an example, we consider the Gaussian integers Z, that is, complex numbers of the form a + bi with a and b in Z. This is an integral domain, and its prime elements are the Gaussian primes. Note that 2 is not a Gaussian prime, because it factors into the product of the two Gaussian primes (1 + i) and (1 − i). The element 3, however, remains prime in the Gaussian integers. In general, rational primes (i.e. prime elements in the ring Z of integers) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.

Related Topics:
Gaussian integer - Complex number - Gaussian prime

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Prime ideals

In ring theory, one generally replaces the notion of number with that of ideal. Prime ideals are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

Related Topics:
Ring theory - Ideal - Prime ideal - Commutative algebra - Algebraic number theory - Algebraic geometry

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The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ...

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A central problem in algebraic number theory is how a prime ideal factors when it is lifted to an extension field. For example, in the Gaussian integer example above, (2) ramifies into a prime power (1 + i and 1 − i generate the same prime ideal), prime ideals of the form (4k + 3) are inert (remain prime), and prime ideals of the form (4k + 1) split (are the product of 2 distinct prime ideals).

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Primes in valuation theory

In class field theory yet another generalization is used. Given an arbitrary field K, one considers valuations on K, certain functions from K to the real numbers R. Every such valuation yields a topology on K, and two valuations are called equivalent if they yield the same topology. A prime of K (sometimes called a place of K) is an equivalence class of valuations. With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.

Related Topics:
Class field theory - Field - Valuation - Topology on ''K'' - Equivalence class - Absolute value - ''p''-adic valuations

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