In abstract algebra , a sub- body of a body is a subset that contains 0 and 1 and, with the restricted links, is itself a body. is then called the upper body of . The pair and is called body enlargement and it is written as or , less often as or .
Definition and spellings
Let be a body, and be a subset of that contains 0 and 1 (the respective neutral elements of the links) and with the limited links addition and multiplication is itself a body. In this case, the lower body (or part of the body ) is called by and the upper body (or extension body ) is called by .
A subset is a subfield of if and only if it contains 0 and 1 and has been completed with regard to the four links addition, multiplication, negation ( i.e. transition from to ) and reciprocal value formation ( i.e. transition from to ), i.e. H. the combination of elements of returns an element of .
The most common spelling for body extensions is (not as a fraction, but next to each other with a slash), sometimes you also find the spelling less often . Some authors just write and add in words that it is a body enlargement.
The spelling corresponds most closely to the phrase " L over K ", but there is little risk of confusion with factor structures such as factor groups or factor spaces , which are also written with a slash.
In a somewhat more general way, one also considers the following case as a body extension: Let , and bodies, partial bodies of and isomorphic to . If it does not lead to misunderstandings and the isomorphism out of context is clear, we can and identify and so even as a subfield of interpret.
A body is called the intermediate body of the body extension if a lower body is from and an upper body is from , so the following applies.
In the following it is always a body expansion.
Degree of expansion
The upper body is a vector space over , where the vector addition is the body addition in and the scalar multiplication is the body multiplication of elements out with elements out . The dimension of this vector space is called and written the degree of expansion . 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 to the complex numbers. The degree of this extension is 2, as a - base of is. In contrast, is (more precisely equal to the thickness of the continuum), so this extension is infinite.
If and are body extensions, then there is also a body extension, and the principle of degree applies
This also applies in the case of infinite extensions (as an equation of cardinal numbers , or alternatively with the usual calculation rules for the symbol infinite ). is called a partial extension of .
Algebraic and transcendent
An element of that is the root of a polynomial over that is not the zero polynomial is called algebraic over . The normalized polynomial of the smallest degree with this zero property is called the minimal polynomial of . If an element is not algebraic, it is said to be transcendent . The case = and = is particularly important. See algebraic number , transcendent number .
If every element is algebraic over , then it is called algebraic extension , otherwise it is called transcendent extension . If every element is transcendent from (that is, from without ), then the extension is called purely transcendent .
One can show that an extension is algebraic if and only if it is the union of all of its finite partial extensions. Hence every finite expansion is algebraic; for example this applies to. Body expansion , on the other hand, is transcendent, if not purely transcendent. But there are also infinite algebraic extensions. Examples are the algebraic closings for the field of rational numbers and for the remainder class fields .
Is a subset of , then the body ( " adjoint ") is defined as the smallest part of the body , which and includes, in other words, the average of all and containing subfield of . consists of all elements of , which can be formed with a finite number of links from the elements of and . If = , then one says, is generated by.
The prime field of a body is the intersection of all subfields of . A prime body is also used to describe a body that has no real partial bodies, i.e. that is itself its own prime body.
Every prime field is isomorphic to the field of rational numbers or one of the remainder class fields (where is a prime number).
If the prime field is isomorphic to , it is said to have zero characteristic . If the prime field is isomorphic to , it is said to have a characteristic .
A body extension that is created by a single element is called simple . A simple extension is finite when it is produced by an algebraic element and purely transcendent when it is produced by a transcendent element. If algebraic, then the degree of expansion is equal to the degree of the minimal polynomial of . A base of is then given by . If, on the other hand, is transcendent, it is isomorphic to the rational function body .
For example, a simple extension of , for with . The expansion cannot be easy as it is neither algebraic nor purely transcendent. Every finite extension of is simple.
More generally, every finite extension of a field with characteristic 0 is a simple extension. This follows from the theorem of the primitive element , which provides a sufficient criterion for simple extensions.
Extension via main ideal rings
Be a principal ideal and any irreducible element of . Then the factor ring is a field, where the main ideal generated by denotes. This theorem can be used to create new upper bodies from bodies with the help of their polynomial ring and to better understand their structure: If a body and an irreducible polynomial of the polynomial ring, then the corresponding upper body (and factor ring is ideal ). Then has in as root so the rest of class, : Substituting for in the remaining Class A, we obtain so is the desired zero in
In general, every finite field can be generated with and prim from the finite field analogous to the following construction of .
Look at the basic body . Then the polynomial irreducible in because it is of degree 2 and has no zero, as can be checked quickly and easily: . has four elements, because division with remainder shows that every remainder class has a unique representative in of degree . Of which there are four: . Since one knows from the above sentence that there is a body and there is only one body with four elements, the following applies .
is not a new equivalence class, because it holds , and since this field has characteristic 2, each element is its own additive inverse (With follows that by subtracting on each side ), so is . This yields by adding : . The multiplication in is inherited as the multiplication of the remainder classes of .
Example: . So the following applies in :
But you can not only define finite upper bodies, but also infinite ones. Here one considers the basic field of the real numbers. Again we need an irreducible polynomial. Probably the best known example of this is . In we now have an infinite number of remainder classes. If one considers this factor ring as a vector space, one can find a basis with two elements . If one now defines , one obtains a 2-dimensional vector space, namely , the field of complex numbers.
With the above sentence one comes to the conclusion that .
Here one can also go over the homomorphism theorem : Define the surjective mapping (for is an archetype). Then , as the smallest polynomial, which has as zero. According to the homomorphism theorem, it also applies here that .
If and are part of the body of , then the smallest common upper body is called the compound of and .
If and both are finally extended upper bodies of , then is also finite.
The decay field of a polynomial is a special field extension.
continue to be a field, a non-constant polynomial over . is a disintegration body of if all zeros of in lie and are minimal in this regard . It is also said that through the adjunction of all roots arises from the beginning. This body is splitting because over into linear factors disintegrates . Every non-constant polynomial has a decay field that is unique except for isomorphism.
For example, has the disintegrating body
In a more general way, the decay field is defined in terms of a set of polynomials: It contains all zeros of all polynomials of this set and is created by the adjunction of all of these zeros . In this case, too, one can prove the existence of a disintegration body that is unambiguous except for isomorphism. If you take over the set of all polynomials , you get the algebraic closure .
is called normal expansion , if all minimal polynomials over from elements in completely decompose into linear factors. If in and its minimal polynomial is over , then the zeros of in are called the algebraic conjugates of . They are exactly the pictures of under -automorphisms of .
A field extension is normal if and only if it is the decay field of a family of polynomials with coefficients from the basic field.
Is not normal about , however, then there is an upper body of that is normal about . It's called the normal shell of .
An example of a non-normal field extension is : The minimal polynomial of the generating element and has complex, not in lying zeros: . Here denote the third root of unity .
A polynomial above is called separable if it has only simple zeros in its decay field. It is separable if and only if it is coprime to its formal derivation . If irreducible, then it is separable if and only if the zero polynomial is not.
But there is also a different definition, according to which a polynomial is called separable if each of its irreducible factors is separable in the above sense. For irreducible polynomials and thus especially for minimal polynomials, the two definitions match, but they differ for reducible polynomials.
An algebraic element of is said to be separable over if its minimal polynomial over is separable. An algebraic extension is called a separable extension if all elements of are separable.
An example for an inseparable field extension is , because the minimal polynomial of the generator breaks down into and thus has a p-fold zero.
However, every extension of a field of characteristic 0 is separable.
Let it be an algebraic closure of . For an algebraic extension , the degree of separability is defined as , the number of homomorphisms from to . For and a minimal polynomial of over is the number of different zeros of in the algebraic closure of . The product formula applies to a tower of algebraic field extensions .
For many fields over which field extensions are examined, irreducible polynomials are always separable and one does not have to worry about the condition of separability for these fields. These bodies are called perfect or perfect .
In a somewhat more formal way, a perfect body can be characterized by one of the following equivalent properties of the body or the polynomial ring :
- Every irreducible polynomial in is separable.
- Every algebraic closure of is a Galois extension (in the broader sense, which is explained in the article Galois group : infinite-dimensional extensions can also be Galois extensions) of .
- Every algebraic field extension of is separable over (and is also perfect again).
- The body either has the characteristic 0 or it has a prime number characteristic and it applies , i.e. i.e., Frobenius endomorphism is bijective.
- The field either has the characteristic 0 or it has a prime number characteristic and every element in has a -th root.
The group of all automorphisms of is called the automorphism group of .
For every automorphism one defines the fixed field of all elements of which are held by. It is easy to calculate that this is a part of the body of . The fixed field (also written as ) of a whole group of automorphisms in is defined by:
The automorphisms of , which leave at least pointwise fixed, form a subgroup of , the group of -automorphisms of , which is also denoted by or .
If the extension is algebraic, normal and separable, then the extension is called Galois ([ ɡaloaːʃ ], after Évariste Galois ). An algebraic extension is Galois if and only if the fixed field of the -automorphism group is equal .
In this case one calls the Galois group of the extension and writes it as , or . In deviation from the terminology used in this article, in the article “Galois group” the group is always referred to as the Galois group , even if the extension is not Galois.
If the Galois group of a Galois extension is Abelian , then this extension is called Abelian ; if it is cyclic, then the extension is called cyclic . For example, is Abelian and Cyclic, because their Galois group is two-element and consists of identity and complex conjugation .
The field of real numbers is - like every real closed or even just Euclidean field more generally - Galois above none of its real partial fields, because the only possible body arrangement there means the identical mapping is the only possible body automorphism.
- is a Galois extension. The automorphism group consists precisely of the identity and the automorphism, which leaves constant but and exchanges. The fixed body of it is .
- is not a Galois extension, because the automorphism group consists only of identity. An automorphism on this extension that is not fixed would have to map to another third root from 2, but does not contain any further third roots from 2. Since it is not a Galois extension, it is also called neither Abelian nor cyclic, although the group ( as a trivial group ) is of course cyclic and Abelian.
- An algebraic closure of any field is Galois if and only if is a perfect field.
The classic problems of ancient mathematics , which deal with the constructibility of a certain number (as a length of the route) using only a compass and ruler from rational numbers, could be reformulated into group-theoretical questions with Galois theory. With the basic idea of René Descartes that the points on straight lines (ruler) and circles (compasses) can be 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 plane or on the basis of already constructed numbers) are exactly the following:
- The rational numbers,
- the square roots of constructible numbers,
- Sum, difference and product of two constructible numbers,
- the reciprocal of any constructible number other than 0.
This shows that every constructible real number
- algebraic and
- is of the degree of a power of two over the field of rational numbers.
This means that for a constructible number the field extension must be a finite, algebraic extension of degree ( ). This is not yet a sufficient condition, but in the classical questions it is sufficient for a proof of impossibility.
- Squaring the circle : Impossible because the circle number is not algebraic.
- Doubling the cube : Impossible: In relation to the constructed starting cube (e.g. a cube with the edge length 1) the new cube would have the edge length . The body enlargement has the degree 3 - no power of two.
- Three-way division of the angle : An angle measuring 60 ° cannot be divided into three equal parts with a compass and ruler. If this angle - i.e. 20 ° - could be constructed, then one could also construct the real number . The addition theorem applies to every angle . So our number solves the equation and is therefore a zero of . Since this polynomial is over irreducible, over has degree 3.
→ The article Euclidean solids shows how a field extension of must be designed so that exactly the numbers that can be constructed with compasses and ruler are present in the extension field.