Update book examples

test/book/cornerstones/algebraic-geometry/alexander-surface.jlcon

@@ -7,7 +7,7 @@ julia> degree(X)
 9
 
 julia> S = ambient_coordinate_ring(X)
-Multivariate polynomial ring in 5 variables over GF(31991) graded by
+Multivariate polynomial ring in 5 variables over GF(31991) graded by 
   x -> [1]
   y -> [1]
   z -> [1]

test/book/cornerstones/algebraic-geometry/circlepar.jlcon

@@ -4,22 +4,22 @@ julia> I = ideal(R, [x^2 + y^2 - 1, y - t*x - 1]);
 
 julia> Gy = groebner_basis(I, ordering = lex([y, x, t]), complete_reduction=true)
 Gröbner basis with elements
-  1 -> x^2*t^2 + x^2 + 2*x*t
-  2 -> y - x*t - 1
+1 -> x^2*t^2 + x^2 + 2*x*t
+2 -> y - x*t - 1
 with respect to the ordering
-  lex([y, x, t])
+lex([y, x, t])
 
 julia> factor(Gy[1])
 1 * (x*t^2 + x + 2*t) * x
 
 julia> Gx = groebner_basis(I, ordering = lex([x, y, t]), complete_reduction=true)
 Gröbner basis with elements
-  1 -> y^2*t^2 + y^2 - 2*y - t^2 + 1
-  2 -> x*t - y + 1
-  3 -> x*y - x + y^2*t - t
-  4 -> x^2 + y^2 - 1
+1 -> y^2*t^2 + y^2 - 2*y - t^2 + 1
+2 -> x*t - y + 1
+3 -> x*y - x + y^2*t - t
+4 -> x^2 + y^2 - 1
 with respect to the ordering
-  lex([x, y, t])
+lex([x, y, t])
 
 julia> factor(Gx[1])
 1 * (y - 1) * (y*t^2 + y + t^2 - 1)

test/book/cornerstones/algebraic-geometry/ex11.jlcon

@@ -4,8 +4,8 @@ julia> I = ideal(R, [x^2+y^2+2*z^2-8, x^2-y^2-z^2+1, x-y+z]);
 
 julia> groebner_basis(I, ordering = lex(R), complete_reduction = true)
 Gröbner basis with elements
-  1 -> 6*z^4 - 18*z^2 + 1
-  2 -> y + 3*z^3 - 9*z
-  3 -> x + 3*z^3 - 8*z
+1 -> 6*z^4 - 18*z^2 + 1
+2 -> y + 3*z^3 - 9*z
+3 -> x + 3*z^3 - 8*z
 with respect to the ordering
-  lex([x, y, z])
+lex([x, y, z])

test/book/cornerstones/algebraic-geometry/ex21a.jlcon

@@ -1,4 +1,4 @@
-julia> J1 = ideal([H(I[1]), H(I[2])])
+julia> J1 = ideal([H(I[1]), H(I[2])]) 
 Ideal generated by
   x0*x2 - x1^2
   x0*x3 - x1*x2

test/book/cornerstones/algebraic-geometry/ex23a.jlcon

@@ -12,11 +12,10 @@ julia> other_pos_abs = pos_abs == 1 ? 2 : 1
 julia> cI2 = cIabs[other_pos_abs][3]
 Ideal generated by
   648*y + (-160*_a^3 - 1269*_a^2 + 22446*_a + 972)*z
-  184147758075888*x + (2850969000960*_a^3 + 22611747888864*_a^2 - 399955313722176*_a - 17319636680832)*y + (2884707374400*_a^3 + 2
-  2782606070410*_a^2 - 405172045313820*_a - 57843867366864)*z
+  184147758075888*x + (2850969000960*_a^3 + 22611747888864*_a^2 - 399955313722176*_a - 17319636680832)*y + (2884707374400*_a^3 + 22782606070410*_a^2 - 405172045313820*_a - 57843867366864)*z
 
 julia> R2 = base_ring(cI2)
-Multivariate polynomial ring in 3 variables over number field graded by
+Multivariate polynomial ring in 3 variables over number field graded by 
   x -> [1]
   y -> [1]
   z -> [1]

test/book/cornerstones/algebraic-geometry/ex314.jlcon

@@ -68,13 +68,11 @@ M1 -> M
 e[1] -> (-x_0*x_3 + x_1*x_2)*e[1]
 Graded module homomorphism of degree [2]
 
-
 julia> phi2 = psi(tohomM1M(hom1[2]))
 M1 -> M
 e[1] -> (-x_0*x_2 + x_1^2)*e[1]
 Graded module homomorphism of degree [2]
 
-
 julia> kerphi2, _ = kernel(phi2);
 
 julia> iszero(kerphi2)

test/book/cornerstones/algebraic-geometry/exres.jlcon

@@ -1,25 +1,25 @@
-julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> S, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
-julia> I = ideal([w^2-x*z,w*x-y*z,x^2-w*y,x*y-z^2,y^2-w*z]);
+julia> J = ideal([w^2-x*z,w*x-y*z,x^2-w*y,x*y-z^2,y^2-w*z]);
 
-julia> A, _ = quo(R, I);
+julia> A, _ = quo(S, J);
 
 julia> FA = free_resolution(A)
 Free resolution of A
-R^1 <---- R^5 <---- R^6 <---- R^2 <---- 0
+S^1 <---- S^5 <---- S^6 <---- S^2 <---- 0
 0         1         2         3         4
 
 julia> FA[1]
-Graded free module R^5([-2]) of rank 5 over R
+Graded free module S^5([-2]) of rank 5 over S
 
 julia> FA[2]
-Graded free module R^5([-3]) + R^1([-4]) of rank 6 over R
+Graded free module S^5([-3]) + S^1([-4]) of rank 6 over S
 
 julia> FA[3]
-Graded free module R^1([-4]) + R^1([-5]) of rank 2 over R
+Graded free module S^1([-4]) + S^1([-5]) of rank 2 over S
 
 julia> map(FA,1)
-R^5 -> R^1
+S^5 -> S^1
 e[1] -> (-w*z + y^2)*e[1]
 e[2] -> (x*y - z^2)*e[1]
 e[3] -> (-w*y + x^2)*e[1]
@@ -28,7 +28,7 @@ e[5] -> (w^2 - x*z)*e[1]
 Homogeneous module homomorphism
 
 julia> map(FA,2)
-R^6 -> R^5
+S^6 -> S^5
 e[1] -> -x*e[1] + y*e[2] - z*e[4]
 e[2] -> w*e[1] - x*e[2] + y*e[3] + z*e[5]
 e[3] -> -w*e[3] + x*e[4] - y*e[5]
@@ -38,7 +38,7 @@ e[6] -> (-w^2 + x*z)*e[1] + (-w*z + y^2)*e[5]
 Homogeneous module homomorphism
 
 julia> map(FA,3)
-R^2 -> R^6
+S^2 -> S^6
 e[1] -> -w*e[2] - y*e[3] + x*e[4] - e[6]
 e[2] -> (-w^2 + x*z)*e[1] + y*z*e[2] + z^2*e[3] - w*y*e[4] + (w*z - y^2)*e[5] + x*e[6]
 Homogeneous module homomorphism

test/book/cornerstones/algebraic-geometry/hilbert-polynomial.jlcon

@@ -1,6 +1,6 @@
-julia> R, (w,x,y,z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> S, (w,x,y,z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
-julia> I = ideal(R, [x*w-y*z, y*w-(x-z)*(x-2*z)]);
+julia> I = ideal(S, [x*w-y*z, y*w-(x-z)*(x-2*z)]);
 
 julia> Q = projective_scheme(I);
 

test/book/cornerstones/algebraic-geometry/param.jlcon

@@ -4,8 +4,7 @@ julia> f = x^5 + 10*x^4*y + 20*x^3*y^2 + 130*x^2*y^3 - 20*x*y^4 + 20*y^5 - 2*x^4
 
 julia> C = plane_curve(f)
 Projective plane curve
-  defined by 0 = x^5 + 10*x^4*y - 2*x^4*z + 20*x^3*y^2 - 40*x^3*y*z + x^3*z^2 + 130*x^2*y^3 - 150*x^2*y^2*z + 30*x^2*y*z^2 - 20*x*
-  y^4 - 90*x*y^3*z + 110*x*y^2*z^2 + 20*y^5 - 40*y^4*z + 20*y^3*z^2
+  defined by 0 = x^5 + 10*x^4*y - 2*x^4*z + 20*x^3*y^2 - 40*x^3*y*z + x^3*z^2 + 130*x^2*y^3 - 150*x^2*y^2*z + 30*x^2*y*z^2 - 20*x*y^4 - 90*x*y^3*z + 110*x*y^2*z^2 + 20*y^5 - 40*y^4*z + 20*y^3*z^2
 
 julia> conics = [x^2-x*z, y^2-y*z];
 

test/book/cornerstones/groups/actions.jlcon

@@ -3,15 +3,14 @@ G-set of
   Alt(4)
   with seeds 1:4
 
-julia> G = dihedral_group(6)
-Pc group of order 6
+julia> G = dihedral_group(6);
 
 julia> U = sub(G, [g for g in gens(G) if order(g) == 2])[1]
-Sub-pc group of order 2
+Pc group of order 2
 
 julia> r = right_cosets(G, U)
 Right cosets of
-  sub-pc group of order 2 in
+  pc group of order 2 in
   pc group of order 6
 
 julia> acting_group(r)
@@ -37,7 +36,7 @@ Group homomorphism
 julia> phi(G[1]), phi(G[2])
 ((2,3), (1,2,3))
 
-julia> function optimal_perm_rep(G)
+julia> function optimal_transitive_perm_rep(G)
          is_natural_symmetric_group(G) && return hom(G,G,gens(G))
          is_natural_alternating_group(G) && return hom(G,G,gens(G))
          cand = []  # pairs (U,h) with U ≤ G and h a map G -> Sym(G/U)
@@ -52,21 +51,21 @@ julia> function optimal_perm_rep(G)
 julia> U = dihedral_group(8)
 Pc group of order 8
 
-julia> optimal_perm_rep(U)
+julia> optimal_transitive_perm_rep(U)
 Group homomorphism
   from pc group of order 8
   to permutation group of degree 4 and order 8
 
 julia> isomorphism(PermGroup, U)
 Group homomorphism
   from pc group of order 8
-  to permutation group of degree 4 and order 8
+  to permutation group of degree 8 and order 8
 
 julia> permutation_group(U)
-Permutation group of degree 4 and order 8
+Permutation group of degree 8 and order 8
 
-julia> for g in all_transitive_groups(degree => 3:9, !is_primitive)
-         h = image(optimal_perm_rep(g))[1]
+julia> for g in all_transitive_groups(degree => 3:8, !is_primitive)
+         h = image(optimal_transitive_perm_rep(g))[1]
          if degree(h) < degree(g)
            id = transitive_group_identification(g)
            id_new = transitive_group_identification(h)
@@ -81,5 +80,3 @@ julia> for g in all_transitive_groups(degree => 3:9, !is_primitive)
 (8, 13) => (6, 6)
 (8, 14) => (4, 5)
 (8, 24) => (6, 11)
-(9, 4) => (6, 5)
-(9, 8) => (6, 9)

test/book/cornerstones/groups/chars.jlcon

@@ -24,7 +24,7 @@ julia> all(i -> s[i] == 0, ord5)
 true
 
 julia> (x -> [x["1a"], x["2b"], x["3a"], x["3b"]]).(filt)
-2-element Vector{Vector{QQAbFieldElem{AbsSimpleNumFieldElem}}}:
+2-element Vector{Vector{QQAbElem{AbsSimpleNumFieldElem}}}:
  [100, 0, 10, 4]
  [1800, 20, 0, 6]
 

test/book/cornerstones/groups/explSL25.jlcon

@@ -4,21 +4,37 @@ SL(2,5)
 julia> T = character_table(G)
 Character table of SL(2,5)
 
-  2  3                 1                 1  3                  1                  1  1  1  2
-  3  1                 .                 .  1                  .                  .  1  1  .
-  5  1                 1                 1  1                  1                  1  .  .  .
-
-    1a               10a               10b 2a                 5a                 5b 3a 6a 4a
-
-X_1  1                 1                 1  1                  1                  1  1  1  1
-X_2  2 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2 -2 -z_5^3 - z_5^2 - 1      z_5^3 + z_5^2 -1  1  .
-X_3  2    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1 -2      z_5^3 + z_5^2 -z_5^3 - z_5^2 - 1 -1  1  .
-X_4  3    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1  3     -z_5^3 - z_5^2  z_5^3 + z_5^2 + 1  .  . -1
-X_5  3 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2  3  z_5^3 + z_5^2 + 1     -z_5^3 - z_5^2  .  . -1
-X_6  4                -1                -1  4                 -1                 -1  1  1  .
-X_7  4                 1                 1 -4                 -1                 -1  1 -1  .
-X_8  5                 .                 .  5                  .                  . -1 -1  1
-X_9  6                -1                -1 -6                  1                  1  .  .  .
+  2  3                 1                 1  3                  1
+  3  1                 .                 .  1                  .
+  5  1                 1                 1  1                  1
+
+    1a               10a               10b 2a                 5a
+
+X_1  1                 1                 1  1                  1
+X_2  2 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2 -2 -z_5^3 - z_5^2 - 1
+X_3  2    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1 -2      z_5^3 + z_5^2
+X_4  3    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1  3     -z_5^3 - z_5^2
+X_5  3 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2  3  z_5^3 + z_5^2 + 1
+X_6  4                -1                -1  4                 -1
+X_7  4                 1                 1 -4                 -1
+X_8  5                 .                 .  5                  .
+X_9  6                -1                -1 -6                  1
+
+  2                  1  1  1  2
+  3                  .  1  1  .
+  5                  1  .  .  .
+
+                    5b 3a 6a 4a
+
+X_1                  1  1  1  1
+X_2      z_5^3 + z_5^2 -1  1  .
+X_3 -z_5^3 - z_5^2 - 1 -1  1  .
+X_4  z_5^3 + z_5^2 + 1  .  . -1
+X_5     -z_5^3 - z_5^2  .  . -1
+X_6                 -1  1  1  .
+X_7                 -1  1 -1  .
+X_8                  . -1 -1  1
+X_9                  1  .  .  .
 
 julia> R = gmodule(T[end])
 G-module for G acting on vector space of dimension 6 over abelian closure of Q
@@ -30,7 +46,7 @@ julia> schur_index(T[end])
 2
 
 julia> gmodule_minimal_field(S)
-G-module for G acting on vector space of dimension 6 over number field
+G-module for G acting on vector space of dimension 6 over number field of degree 4 over QQ
 
 julia> B, mB = relative_brauer_group(base_ring(S), character_field(S));
 

test/book/cornerstones/groups/intro.jlcon

@@ -62,18 +62,12 @@ julia> pts = collect(orb)
 
 julia> visualize(convex_hull(pts))
 
-julia> R2 = free_module(K, 2) # the "euclidean" plane over K
-Vector space of dimension 2 over QQBar
-
-julia> A = R2([0,1])
-(Root 0 of x, Root 1.00000 of x - 1)
-
-julia> pts = [A*mat_rot^i for i in 0:4];
+julia> R2 = vector_space(K, 2); # the "euclidean" plane over K
 
 julia> sigma_1 = hom(R2, R2, [-R2[1], R2[2]])
 Module homomorphism
-  from vector space of dimension 2 over QQBar
-  to vector space of dimension 2 over QQBar
+  from vector space of dimension 2 over field of algebraic numbers
+  to vector space of dimension 2 over field of algebraic numbers
 
 julia> rot = hom(R2, R2, mat_rot);
 
@@ -127,10 +121,10 @@ julia> describe(H)
 "a finitely presented group"
 
 julia> H1, _ = derived_subgroup(H)
-(Sub-finitely presented group, Hom: H1 -> H)
+(Finitely presented group, Hom: H1 -> H)
 
 julia> H2, _ = derived_subgroup(H1)
-(Sub-finitely presented group, Hom: H2 -> H1)
+(Finitely presented group, Hom: H2 -> H1)
 
 julia> describe(H2)
 "Z x Z"

test/book/cornerstones/groups/reps.jlcon

@@ -12,9 +12,9 @@ julia> M = regular_gmodule(GF(7), G);
 
 julia> C = composition_factors_with_multiplicity(M)
 3-element Vector{Any}:
- (G-module for G acting on vector space of dimension 1 over GF(7), 1)
- (G-module for G acting on vector space of dimension 1 over GF(7), 1)
- (G-module for G acting on vector space of dimension 4 over GF(7), 2)
+ (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 4 over prime field of characteristic 7, 2)
 
 julia> [is_absolutely_irreducible(x[1]) for x in C]
 3-element Vector{Bool}:
@@ -28,12 +28,12 @@ Morphism of finite fields
   to finite field of degree 2 and characteristic 7
 
 julia> M = extension_of_scalars(C[3][1], phi)
-G-module for G acting on vector space of dimension 4 over GF(7, 2)
+G-module for G acting on vector space of dimension 4 over finite field of degree 2 and characteristic 7
 
 julia> composition_factors_with_multiplicity(M)
 2-element Vector{Any}:
- (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
- (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
+ (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
 
 julia> G = pc_group(symmetric_group(4));
 
@@ -63,15 +63,15 @@ julia> pushfirst!(im, hom(FF, FF, matrix(C, [0 1; 1 0])));
 julia> phi = gmodule(G, im);
 
 julia> character(phi)
-class_function(character table of G, [2, 0, -1, 2, 0])
+class_function(character table of G, QQAbElem{AbsSimpleNumFieldElem}[2, 0, -1, 2, 0])
 
 julia> schur_index(ans)
 1
 
 julia> T = matrix(C, [C(1) -z; z+1 z]);
 
 julia> [matrix(x)^T for x in action(phi)]
-4-element Vector{AbstractAlgebra.Generic.MatSpaceElem{QQAbFieldElem{AbsSimpleNumFieldElem}}}:
+4-element Vector{AbstractAlgebra.Generic.MatSpaceElem{QQAbElem{AbsSimpleNumFieldElem}}}:
  [1 0; -1 -1]
  [0 1; -1 -1]
  [1 0; 0 1]

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

@@ -55,18 +55,18 @@ 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> b = preimage(f, c[1]) # the element might different per session
-a^3
+a
 
 julia> A, mA = automorphism_group(K);
 
 julia> X = matrix(ZZ, [coordinates(OK(mA(s)(b))) for s in A])
+[0    1    0    0]
 [0    0    0    1]
 [1   -1    1   -1]
 [0    0   -1    0]
-[0    1    0    0]
 
 julia> det(X)
--1
+1
 
 julia> k, = number_field(x^4 - 13*x^2 + 16); candidates = []; for F in abelian_normal_extensions(k, [2], ZZ(10)^13)
          K, = absolute_simple_field(number_field(F))