Katsura documentation (oscar-system#2594)

* Fixes description of katsura example, generates katsura(n) over QQ

* adds higher level descirption of katsura-n examples in documentation

docs/src/CommutativeAlgebra/ideals.md

@@ -262,6 +262,14 @@ dehomogenization(F::MPolyDecRingElem, pos::Int)
 
 ## Generating Special Ideals
 
+### Katsura-n
+
+These systems appeared in a problem of magnetism in physics.
+For a given $n$ `katsura(n)` has $2^n$ solutions and is defined in a
+polynomial ring with $n+1$ variables over the rational numbers. For a
+given polynomial ring `R` with $n$ variables `katsura(R)` defines the
+corresponding system with $2^{n-1}$ solutions.
+
 ```@docs
 katsura(n::Int)
 ```

src/Rings/special_ideals.jl

@@ -13,20 +13,23 @@ end
 @doc raw"""
     katsura(n::Int)
 
-Given a natural number `n` return the Katsura ideal generated by
-$u_m - \sum_{l=n}^n u_{l-m} u_l$, $1 - \sum_{l = -n}^n u_l$
+Given a natural number `n` return the Katsura ideal over the rational numbers generated by
+$u_m - \sum_{l=-n}^n u_{l-m} u_l$, $1 - \sum_{l = -n}^n u_l$
 where $u_{-i} = u_i$, and $u_i = 0$ for $i > n$ and $m \in \{-n, \ldots, n\}$.
 
 Note that indices have been shifted to start from 1.
 
 # Examples
 ```jldoctest
-julia> katsura(2)
+julia> I = katsura(2)
 ideal(x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2, 2*x1*x2 + 2*x2*x3 - x2)
+julia> base_ring(I)
+Multivariate polynomial ring in 3 variables x1, x2, x3
+  over rational field
 ```
 """
 function katsura(n::Int)
-    R, _ = polynomial_ring(ZZ, n + 1)
+    R, _ = polynomial_ring(QQ, n + 1)
     return katsura(R)
 end