Finite field
In algebra, a branch of mathematics, a finite field or Galois field is a set with a finite number of elements on which the basic operations of addition and multiplication are defined and satisfies all the properties of a body. In honor of Évariste Galois, who has already anticipated some imaginary numbers modulo p, Eliakim Hastings Moore probably coined in 1893 the English term Galois field.
The set of Wedderburn states that multiplication in a finite division ring is commutative necessary. This means that finite division ring are always finite fields.
Exists for each prime and each positive natural number ( up to isomorphism ) exactly one body with elements, called with or. the body of the residue classes of integers modulo.
Finite fields play an important role in cryptography and coding theory ( forward error correction, such as Reed -Solomon code). In addition, they are fundamental to the study of prime ideals in the ring of integers of a finite field extension of the framework of algebraic number theory. Compare this to the section on Dedekind rings in the article " branching ".
In addition, finite fields are in geometry as coordinate spaces of finite geometries of importance. They are more general coordinate ranges of levels and spaces in synthetic geometry. Using the addition and multiplication in a finite field for links with weaker algebraic properties are defined, consisting of the body, for example, make a Ternärkörper or quasi body. This generalized projective then bodies and affinity levels can be constructed.
Example: The body with 2 elements
The residue classes modulo 2 forming the body with two elements. representing the residue class of even numbers, the residue class of odd numbers. For the addition of the following applies:
For multiplication applies:
Classification of finite fields
The core of the homomorphism is a finite field, it is always of the form, that it consists of all multiples of a certain prime number. In this case, it is noted that one is not a prime. This prime number is called the characteristic of. The image of is isomorphic to the residue field and is called the prime field. As a finite extension field is also an -dimensional vector space over its prime field. Thus has exactly elements.
In ( a body ) characteristic is a homomorphism of additive groups, because
The remaining occurring on the right side of the binomial formula summands fall away because of. contributes to honor Ferdinand Georg Frobenius ' the name Frobeniushomomorphismus. The prime field is fixed by pointwise (in fact, for example, is a multiple of 7). Just as every body with elements. On the other hand, as a polynomial of degree at most distinct zeros. These are all captured by the elements of.
From this it can be concluded:
- There are up to isomorphism exactly one body with elements. ( For each prime and each natural number)
- This represents a Galois extension of its prime field
- The Galois group is cyclic of order and is generated from.
Other properties of finite fields:
- All elements except 0, the additive group of a finite field of characteristic have order
- As with any finite separable field extension, there is always a primitive element, ie one such that the extension field only this arises by adjoining an element. If the minimal polynomial of, so has the degree and. Furthermore, it is always already the splitting of, ie decays via already completely into linear factors.
- Is a divisor of, as a Galois extension of degree. The corresponding Galois group is also cyclic and is generated by the th power of the Frobenius homomorphism.
Multiplicative group and discrete logarithm
The multiplicative group of the finite field is composed of all elements of the body with the exception of zero. The group operation is multiplication of the body.
The multiplicative group is a cyclic group with elements. Since therefore applies to all elements of this group, every element is a root of unity of the body. Those roots of unity, the generator of the multiplicative group are called primitive roots of unity or primitive roots. These are the distinct zeros of th Kreisteilungspolynoms. ( Denotes the Euler φ - function. )
Is a primitive root of the multiplicative group, then the multiplicative group can be represented as a set. Such is therefore also known as a producer or generator. For each element there with a uniquely determined number. This number is called discrete logarithm of to the base. Although can be calculated easily for each, the task is to find for given the discrete logarithm, according to current knowledge, for large numbers is an extremely computationally expensive operation. Therefore, the discrete logarithm is used in cryptography, such as the Diffie-Hellman key exchange.
Other examples
The body of 49 elements
The prime field is -1 not a square. This follows from the first additional set for quadratic reciprocity law of Carl Friedrich Gauss or - at such a small prime - by explicitly squaring of all six elements of the multiplicative group. Just as the complex numbers from the real numbers by adjoining arise from, can also win by adjoining a " number"; formally correct than the same time is also a factor ring of the ring of Gaussian integers.
The body of 25 elements
In characteristic 5 -1 is always a square. However, none of squares modulo 5 are the numbers 2 and 3 (in characteristic with are always exactly half of the elements of the multiplicative group squares or non- squares. ) So you can receive the body with 25 elements as, so by adjunction of.
For the historical development
That one with numbers modulo a prime " as rational numbers " can be expected, Gauss had already shown. Galois introduced the bill modulo p imaginary number sizes, just like the imaginary unit in the complex numbers. He has probably seen as the first field extensions of - even if the abstract body term was introduced in 1895 by Heinrich Weber and Frobenius first 1896 this extended to finite structures. In addition, Eliakim Hastings Moore and previously appeared in 1893 already studied finite fields and introduced the name Galois field.