# Für Wolfram

## Docstring für `PolynomialRing`

Wir haben den docstring hinzugefügt, aber es wird noch ein paar Tage dauern bis er in Oscar auftaucht.

## Types of base fields

- The field of rational numbers $\mathbb{Q}$:

  julia> QQ
  Rational Field

- Finite fields $\mathbb{F_p}$, $p$ a prime:

   julia> GF(3)
   Galois field with characteristic 3

   julia> GF(ZZ(2)^127 - 1)
   Galois field with characteristic 170141183460469231731687303715884105727

- Finite fields $\mathbb{F}_{p^n}$ with $p^n$ elements, $p$ a prime.

   julia> FiniteField(2, 70, "a")
   (Finite field of degree 70 over F_2, a)

- Transcendental extension of $\mathbb{Q}$ or $\mathbb{F}_p$.

    julia> Qx, x = PolynomialRing(QQ, 5, "x")
    (Multivariate Polynomial Ring in x1, x2, x3, x4, x5 over Rational Field, fmpq_mpoly[x1, x2, x3, x4, x5])

    julia> F = FractionField(Qx)
    Fraction field of Multivariate Polynomial Ring in x1, x2, x3, x4, x5 over Rational Field

- Simple algebraic extensions of $\mathbb{Q}$ or $\mathbb{F}_p$:

   julia> Qx, x = QQ["x"]
   (Univariate Polynomial Ring in x over Rational Field, x)

   julia> K, a = NumberField(x^2 + 1, "a")
   (Number field over Rational Field with defining polynomial x^2 + 1, a)
   
   julia> F = GF(3)
   Galois field with characteristic 3

   julia> Fx, x = F["x"]
   (Univariate Polynomial Ring in x over Galois field with characteristic 3, x)

   julia> K, a = FiniteField(x^2 + 1, "a")
   (Finite field of degree 2 over F_3, a)

- The field of real numbers represented by floating point numbers of a user defined precision, the field of complex numbers represented by (pairs of) floating point numbers of a user defined precision.
*Wir haben Körper von reellen und komplexen Zahlen, die durch Intervallarithmetik implementiert sind. Wenn wir das andere auch wollen, müssten wir da noch was machen. Andererseits weiß ich nicht wie korrekt da die Gröbnerbasen überhaupt sind und ob das ganze Sinn ergibt.*

  julia> RR = Nemo.RealField(64)
  Real Field with 64 bits of precision and error bounds

  julia> a = RR(2.1)
  [2.1000000000000000888 +/- 1.79e-20]

  julia> inv(a)
  [0.4761904761904761703 +/- 5.78e-20]

  julia> CC = Nemo.ComplexField(64)
  Complex Field with 64 bits of precision and error bounds

  julia> a = CC(-2.1)
  -[2.1000000000000000888 +/- 1.79e-20]

  julia> sqrt(a)
  [1.449137674618943888 +/- 1.26e-19]*im

- The ring of integers $\mathbb{ZZ}$

  julia> ZZ
  Integer Ring

- Residue rings of $\mathbb{Z}$

  julia> ResidueRing(ZZ, 20)
  Integers modulo 20

## Hier noch der andere Konstruktor. Hoffe das ist was du im Sinn hattest:

julia> Qx, x = PolynomialRing(QQ, 2, "x");

julia> x
2-element Vector{fmpq_mpoly}:
 x1
 x2

*(Ich glaube die Funktion hat keinen docstring)*

## Hier noch ein paar Konstruktoren, auf die ich besonders stolz bin:

julia> Qx, x = @PolynomialRing(QQ, x[1:3]);

julia> x
3-element Vector{fmpq_mpoly}:
 x[1]
 x[2]
 x[3]

julia> Qx, x = @PolynomialRing(QQ, x[1:3, 1:3]);

julia> x
3×3 Matrix{fmpq_mpoly}:
 x[1,1]  x[1,2]  x[1,3]
 x[2,1]  x[2,2]  x[2,3]
 x[3,1]  x[3,2]  x[3,3]

julia> Qx, x, y = @PolynomialRing(QQ, x[1:3], y[1:2]);

julia> x, y
(fmpq_mpoly[x[1], x[2], x[3]], fmpq_mpoly[y[1], y[2]])

julia> I = [1, 3, 5];

julia> Qx, x = @PolynomialRing(QQ, x[I]);

julia> x
3-element Vector{fmpq_mpoly}:
 x[1]
 x[3]
 x[5]