Another batch of book test changes

test/book/cornerstones/number-theory/galoismod.jlcon

@@ -52,9 +52,9 @@ true
 
 julia> V, f = galois_module(K); OK = ring_of_integers(K); M = f(OK);
 
-julia> fl, c = is_free_with_basis(M); # the elements of c are a basis
+julia> fl, c = is_free_with_basis(M); # the elements of c form a basis
 
-julia> b = preimage(f, c[1]) # the element might different per session
+julia> b = preimage(f, c[1]) # the element may be different per session
 a
 
 julia> A, mA = automorphism_group(K);

test/book/specialized/bies-turner-string-theory-applications/SU5-2.jlcon

@@ -11,18 +11,3 @@ Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based
 
 julia> t5 = resolve(t, 1)
 Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)
-
-julia> cox_ring(ambient_space(t5))
-Multivariate polynomial ring in 12 variables over QQ graded by
-  x1 -> [1 0 0 0 0 0 0]
-  x2 -> [0 1 0 0 0 0 0]
-  x3 -> [0 1 0 0 0 0 0]
-  x4 -> [0 1 0 0 0 0 0]
-  x -> [0 0 1 0 0 0 0]
-  y -> [0 0 0 1 0 0 0]
-  z -> [0 0 0 0 1 0 0]
-  e1 -> [0 0 0 0 0 1 0]
-  e4 -> [0 0 0 0 0 0 1]
-  e2 -> [-1 -3 -1 1 -1 -1 0]
-  e3 -> [0 4 1 -1 1 0 -1]
-  s -> [2 6 -1 0 2 1 1]

test/book/specialized/decker-schmitt-invariant-theory/H3.jlcon

@@ -47,22 +47,22 @@ julia> [[ kernel(reduce(hcat, [ K[coeff(p, x); coeff(p, y); coeff(p, z)] for p i
 1-element Vector{Vector{AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}}}:
  [[a+1 1 0], [-a 1 0], [-1 1 0], [a+1 0 1], [-a 0 1], [-1 0 1], [0 -a 1], [0 a+1 1], [0 -1 1]]
 
-julia> T, t = polynomial_ring(QQ, "t")
-(Univariate polynomial ring in t over QQ, t)
+julia> S, s = QQ["t"]; T = fraction_field(S); t = T(s);
 
 julia> RR, (X, Y, Z) = graded_polynomial_ring(T, [ "X", "Y", "Z"]);
 
 julia> F = X^3+Y^3+Z^3;
 
-julia> G = t*F+hessian(F);
+julia> G = t*F+hessian(F)
+t*X^3 + 216*X*Y*Z + t*Y^3 + t*Z^3
 
 julia> L = syzygy_generators([hessian(G), F, hessian(F)]);
 
 julia> collect(coefficients(L[1]))
-3-element Vector{QQPolyRingElem}:
- 1
- 279936*t
- -t^3 - 93312
+3-element Vector{AbstractAlgebra.Generic.FracFieldElem{QQPolyRingElem}}:
+ -1
+ -279936*t
+ t^3 + 93312
 
 julia> C, iC = center(H3);