Quadratic field

A quadratic number field is an algebraic field extension of the form

With a rational number which is not a square in. These are precisely the extensions of degree over

Quadratic number fields are, by themselves, the simplest number field.

Introduction

The theory of quadratic number fields developed from the study of binary quadratic forms. Euler and Fermat had gathered in their investigations to Diophantine equations, many fundamental individual results, which then left room for further research. In his Disquisitiones Arithmeticae Gauss shall, in Section V on from the work of Fermat, Euler, and Lagrange, and there treated extensively the theory of binary quadratic forms. Although Gauss moved with his representation in the domain of integers, it is more elegant from today's perspective, to expand the field of rational numbers as square that a decomposition of quadratic forms can be made into linear factors. Such a decomposition looks for example like this:

Thus, the theory of the quadratic component of the body to a theory of the binary square shapes.

The field of rational numbers can be extended in various ways to form a comprehensive body. So you studied about the ring of algebraic integers. It contains exactly those complex numbers, the root of a normalized polynomial with integer coefficients. It is in an expansion often useful to take only as many numbers as needed for a given problem:

Be a finite number of algebraic numbers and be the smallest subfield of the field of algebraic numbers, which contains all these numbers. Then one writes

And says that the body is an extension field of the created by adjoining the elements. The couple and are referred to as field extension and writes for

In particular, is an abelian group. Moreover, because the multiplication of elements of the scalars via

Is explained, is obtained from the body of axioms directly the vector space axioms, so that may be construed as a vector space over. The body has over finite degree, that is, as a vector space is finite-dimensional.

Is produced by an algebraic number, then has a base, and thus the dimension

Being equal to the degree of the minimal polynomial that has as root. It can be shown that the degree has over 2, if the minimal polynomial of is square. Thus, a quadratic number field.

Called for a number field

The wholeness of the ring or the whole concluding in thus consists of all elements that are in integral-algebraic; That is, it holds:

Definition

A quadratic number field is a quadratic extension of the rational numbers. Quadratic number fields arise from so by adjoining the square root.

Be below one of 0 and 1 different square-free integer. Then, ie, the amount

A quadratic number field.

Is so called real quadratic number fields, otherwise imaginary quadratic number field. This is an arbitrary but fixed chosen complex solution of the equation. The second solution of this equation leads to the same number field.

Properties

Konjugationsabbildung

It is considered that each element of the zero of a polynomial of degree. So every element is algebraic of. One thus obtains a tower of bodies:

In particular, a basis from which it is called

Now the body has exactly two Körperautomorphismen, firstly, the identity map

And on the other Konjugationsabbildung:

In particular, a Galois group of order 2 is called For the conjugated element.

Norm and trace

The two sizes norm and trace of a quadratic number field can be by means of its Körperautomorphismus represented as follows:

And

Since the embedding forms a ring homomorphism, the multiplicative norm and the trace is additive. Substituting we get:

The standard is therefore a square shape due to the fact that the algebraic integers form a ring, is obviously also a ring. This assumes an analogous role in how the ring in and it is so is a subring of means that all elements of the form always integral-algebraic, and one obtains an inclusion of rings:

That does not necessarily equality holds here, shows the following

There is a very easy way to identify the algebraic integers in a quadratic number field, because a number is exactly then when its norm and trace are integers.

There is countably infinite, is also countably infinite, since each has only finitely many zeros. Therefore, the set of algebraic numbers is countably infinite.

There remains the question of the form of the integral algebraic elements of this case, the many variations of the elements and of the congruence modulo 4 depend. As a square-free number can be congruent modulo 4 from the outset only to 1, 2 or 3. It now applies:

Units

A first essential difference between real and imaginary quadratic number fields is in terms of their units. For example, is the unit group of the ring, the cyclic group of order The description of the unit group of the ring wholeness however, depends on whether is real or imaginärquadratisch. Thus, the unit group for imaginärquadratische number field is finite and we can describe as follows:

In case of a real quadratic number field, the description of the unit group is more complex. It is shown that every real quadratic number field has infinitely many units. The determination of the unit group boils down to the solution of the Pell equation. It can be shown by means of the Dirichlet drawer principle that this equation provides an infinite number of units ( solutions). As the drawer is not constructive principle, is used for the determination of the units, the continued fraction expansion of

Construction of quadratic number fields

A classic example of the construction of a quadratic number field to take the uniquely determined quadratic intermediate body of a cyclotomic field formed by a primitive -th root of unity, is an odd prime. Uniqueness, it follows that the Galois group of isomorphic to and thus is cyclic. By looking at the branch can be seen that the square is equal to the intermediate body; the discriminant of is in fact a potency, and therefore this must also apply to the discriminant of the quadratic intermediate body. According to the above statement, therefore, must be, otherwise there is branched. The same also applies to any of magnitude to an odd prime number.

The body has, however, just the three bodies, and as a quadratic intermediate body; This is because that the Galois group of the extension no longer cyclically (see prime residue class group).

For the special case we obtain the whole ring of Gaussian integers, for the whole ring of Eisenstein numbers. These two rings are the only wholeness wholeness rings quadratic number fields, which are also cyclotomic fields.

Non-uniqueness of the prime factorization

In 1843, Peter Ernst Eduard Kummer Dirichlet drew attention to the non-uniqueness of the prime factorization in certain number of rings. Kummer had, the fundamental theorem of number theory for all algebraic numbers regarded by his supposed proof to Fermat 's conjecture, which integrated the algebraic numbers as proven, so they also have a unique decomposition as the ordinary integers. That this is not more already given in the ring, can easily be shown for the number 21.

Thus, on the one hand and on the other hand. The fact that the numbers are irreducible and not associated with each other in all, you can see using the standard as follows. Suppose the number 3 would be dismantled. Some with, and no units were. Then and therefore need to be. Now, with the mold and hence it follows that the norm. Now the equation is obviously unsolvable in the integers, which is in contradiction to our assumption. So the number is irreducible in and you proved analogously that there are also the numbers. The fact that the numbers and not associated to each other, is clear. Just as can be and not associated as conjugate to each other. Suppose the numbers and had to be associated, then would the breaks. However, since both the track of as well as are not integers, the elements can not be in thus. So the numbers are not associated with each other. Consequently, there are two different prime factorizations for the number in front.

So we see that the fundamental theorem of number theory and thus the uniqueness of prime factorization in general can no longer be assumed.

Problems of this kind are to get to grips with today, the Kummer's ideal theory. Guided by the complex numbers was grief intention is to provide an extended range of new ideal numbers, so that they can be broken down into the product ideal prime numbers clearly. Developed by Kummer theory of ideal numbers was systematized by the German mathematician Richard Dedekind and today is called the ideal numbers simply as the Dedekind ideals of the ring. The fundamental theorem of Dedekind's ideal theory now provides the generalization of the theorem of unique factorization and shows a way to deal with the ambiguity of the prime factorization, and an analogy to the Fundamental Theorem of number theory to be restored. (See about Dedekind ring).

Prime ideal

The fact that the prime ideal of a principal ideal, for a prime number, can not be arbitrary, it follows, from the standard. This means decays into either a prime ideal or the product of two (not necessarily different ) prime ideals of norm. An odd prime number is called in

• Sluggish when a prime ideal,

• Decomposed when with prime ideals,

• Branched when. a prime ideal

The third case occurs right at the ( finitely many ) prime divisors of the discriminant. The other two cases occur in a certain sense " equally often " on; this follows from the tightness Chebotarevschen sentence.

One can find now without much effort, that for the discriminant of a quadratic field:

Note that always applies.

With the help of the discriminant and the Legendre symbol can be a clear description of the behavior of odd primes in a quadratic number field enter:

Is, then, and is branched,

If, then is disassembled,

If, then is sluggish.

Proof: See: decomposition law

Note: The prime was excluded. It is true, however, that is sluggish when. It is disassembled, when, and it is branched, optionally.

The statement for the inertia also applies to the decomposition into prime elements; in general, such statements can but just then prime elements continued when principal ideal ring, thus has unique decomposition into prime elements, or equivalently has class number.

Consider one example. Then we get by repeated application of the quadratic reciprocity law that the prime in is sluggish. Because.