Field extension
In abstract algebra, a subfield K of a body is a subset containing 0 and 1 and with the on restricted links is a body itself. is called by then upper body. The couple and are referred to as field extension and writes it as or rarer than or.
For example, the field of complex numbers is a torso of the body of the real numbers and therefore a field extension.
Definition and notation
Is a body, and is a subset of which contains 0 and 1 ( the corresponding neutral elements of the links ) and restricted threads addition and multiplication with the even one body. In this case, is called lower body (or body part ) of and is called upper body (or body extension ) of.
A subset is exactly then a subfield of, if it contains 0 and 1 and with respect to the four shortcuts addition, multiplication, negation (ie transition from to ) and inverse formation is complete (ie transition from to ), that is, the combination of elements of supplies again an element of.
The most common notation for field extensions (not as a fraction, but side by side with a slash ), sometimes you will also find, rare spelling. Some authors write simply and add in words that there is a field extension.
The notation corresponds most closely to the speech "L over K ", but there is a slight confusion with factor structures such as factor groups or factor spaces that are also written with a slash.
More generally, one considers also the following case as a field extension: Let, and body, body part by and isomorphic to. If it does not lead to misunderstanding and the isomorphism from the context is clear, and you can identify and so even regarded as part of the body.
A body called intermediate body, the body extension if a lower body and an upper body is so true.
It should be in the following always a field extension.
Extension degree
The upper body is a vector space over, with the vector addition is the body -addition and scalar multiplication in the body multiplication of elements with elements from. The dimension of this vector space is called and written degree of extension. The extension is called finite or infinite, depending on whether the degree is finite or infinite.
An example of a finite field extension is the extension of the real numbers of the complex number. The degree of this extension is 2 because a base is. In contrast, (more precisely equal to the cardinality of the continuum ), so this expansion is infinite.
Are and field extensions, then is also a field extension, and it is the degree set
This also applies in the case of infinite extensions ( as an equation of cardinal numbers, or alternatively with the usual rules for computing the symbol infinity). this is called a partial extension of.
Algebraic and transcendental
Is an element of L, the zero of a polynomial over K, is called algebraic over K. The normalized polynomial of least degree with this property is called minimal polynomial of zeros. Is not algebraic one element, then it is called transcendent. The case L = K = and is particularly important. See algebraic number, transcendental number.
If every element of L algebraic over K, then that L / K algebraic extension, otherwise transcendental extension. If every element of L \ K is transcendent ( ie from L without K), it means the extension of a purely transcendental.
One can show that an extension is algebraically if and only if it is the union of all its finite part extensions. Thus any finite extension is algebraic; For example, this applies to. The field extension, however, is transcendent, if not purely transcendent. But there are also infinite algebraic extensions. Examples are the algebraic statements for the field of rational numbers, and the residue class field.
Körperadjunktion
V is a subset of L, then the body is C ( V) ( "K adjoint V " ) is defined as the smallest part of the body of L, containing K and V, containing, in other words, the average of all K and V subfield of L. K (V) consists of all the elements of L, which may be formed with a finite number of connections of the elements of K and V. L = K (V), then said to L is generated by V.
Prime field
The prime field of a body K is the average of all subfield of K. As a prime field is also referred to a body K, which has no proper subfield, that is his own prime field itself.
Each prime field is the field of rational numbers or the residue field isomorphic (where a prime number ).
If the prime field of K is isomorphic to, then we say that K has characteristic zero. Is the prime field of K is isomorphic to, we say K have characteristic.
Easy Expansion
A field extension generated by a single element is called simple. A simple extension is finite, if it is generated from an algebraic element and transcendental pure when it is generated by a transcendental element.
For example, a simple extension of, because with. The extension may not be easy, since it is neither algebraic nor purely transcendent. Every finite extension is simple.
More generally, any finite extension of a body with characteristic 0 is a simple extension. This follows from the theorem of the primitive element, which provides a sufficient criterion for simple extensions.
Compound
Sind and part of body, then that means the lowest common upper body of the compound and.
Are both finite and extended upper body, then it is also finite.
Splitting field
The splitting field of a polynomial is a special field extension.
K was further includes a body, a non-constant polynomial over K. L / K is a splitting field of p if all the zeros of p lie in L and L is minimal in this respect. We also say that L is formed by adjoining all roots of p in K. This body is called splitting, since p splits over L into linear factors. Every non-constant polynomial has a unique up to isomorphism splitting.
For example, the splitting field
More generally we define the splitting with respect to a set of polynomials: This contains all the zeros of all polynomials of this set and created by adjoining all these zeros at. Also in this case, one can prove the existence of a unique up to isomorphism splitting field. If we take the set of all polynomials over, we obtain the algebraic degree.
Normal extensions
L / K is called normal extension if all the minimal polynomials over K of elements of L in L disintegrate completely into linear factors. If a is in L and f be minimal polynomial over K, then the zeros of f in L are called the conjugates of a, you are exactly the images of a under K- automorphisms of L.
A field extension is normal if and only if it is a splitting field of a family of polynomials with coefficients in the base.
If L is not normal over K, then there is, however, an upper body of L, which is normal over K. It is called the normal envelope of L / K.
An example of a non-normal field extension with: the minimal polynomial of the generating element is complex and has, thus not lying in L, zeros.
Separability
Separable polynomials
A polynomial f over K is called separable if it has only simple zeros in its splitting field. It is precisely then separable when it comes to its formal derivative f ' is prime. If f is irreducible, then it is separable if and only if f ' is not the zero polynomial.
But there is also a different definition, according to which a separable polynomial is, if each of its irreducible divisor is separable in the above sense. For irreducible polynomials and thus in particular for minimal polynomials both definitions agree for reducible polynomials they differ, however.
Separable extensions
An algebraic element of is called separable over if its minimal polynomial is separable. An algebraic extension is called separable extension if all the elements of are separable.
An example of an inseparable field extension is because the minimal polynomial of the generator is divided into, and thus has a p- multiple zero.
The Separabilitätsgrad an algebraic field extension is defined as the number of homomorphisms of - the -containing algebraic completion of which are the identity. For and a minimal polynomial of over is the number of distinct zeros of the algebraic degree of. For a tower of algebraic field extensions of the product formula holds.
Perfect body
For many body over which body extensions are examined, irreducible polynomials are always separable and one must not in these bodies look after the condition of separability. One calls this body completely or perfectly.
More formally, a perfect body are characterized by one of the following equivalent properties of the body or of the polynomial ring:
Particular field of characteristic 0, finite fields and algebraically closed fields are perfect. An example of a perfect body is not - there is the body element X is not p- th root.
K- automorphisms
The group of all automorphisms of is called the automorphism group of.
For each automorphism defining the fixed field of all elements of L that are held by. It is easy to check that this is a subfield of L. The fixed field (also written as ) a whole group G of automorphisms in L is defined by:
The automorphisms of L which leave pointwise fixed at least K, form a subgroup of Aut (L ), the group of K- automorphisms of L, which is denoted by or.
Galois extension
Galois groups
If the extension L / K algebraic, normal and separable, then is called the Galois extension ( [ ɡaloa ː ʃ ], after Évariste Galois ). An algebraic extension is Galois if and only if the fixed field Fix ( Aut ( L / K )) of K- automorphism is equal to K.
It's called Aut ( L / K) in this case, the Galois group of the extension and writes it as, or. Notwithstanding the used in the present article language regime in the article " Galois group ", the group Aut ( L / K ) is always referred to as Galois group, even if the extension L / K is not Galois.
If the Galois group of a Galois extension abelian, then this is called abelian extension, then it is cyclic, then that means the expansion cycle. For example, is abelian and cyclical as their Galois group is zweielementig and consists of the identity and complex conjugation.
The field of real numbers is - as any general real closed or even Euclidean body - more than any of its proper subfield Galois, because there the only possible body configuration is the identity map is the only possible Körperautomorphismus.
Examples
- Is a Galois extension. The automorphism group consists exactly of the identity and the automorphism which can be constant, however, and reversed. The fixed field thereof.
- Is not a Galois extension, because the automorphism group A consists only of the identity. An automorphism of this extension, which can not fix would have to be mapped to a different cube root of 2, but contains no other cube roots of 2, since it is not a Galois extension, it also means neither abelian nor cyclical, although the group A (as trivial group) course is cyclic and abelian.
- An algebraic closure of an arbitrary body is just over a Galois field then, if a perfect body.
Konstruierbarkeitsfragen
The classic problems of ancient mathematics, which are (as length) is about the constructability of a certain number alone with compass and ruler of rational numbers could be reformulated with the Galois theory in group-theoretical questions. With the philosophy of René Descartes that the points on the straight line ( ruler) and circles ( circles ) are represented by analytical equations, it can be shown that the constructible numbers (coordinates of finite intersections of two of these figures in the rational number level or at the already constructed base numbers) are exactly the following:
- The rational numbers,
- The square roots of constructible numbers,
- Sum, difference and product of two constructible numbers,
- The inverse of each other than 0 constructible number.
Thus one can show that every constructible real number
This means that for a number c constructible body extension must be a finite algebraic extension of degree (). This is not a sufficient condition, but suffice in the classic questions for an impossibility proof.
→ In the article Euclidean body is represented as a field extension must be of fundraising so that the constructible with compass and straightedge numbers in the extension field are exactly available.