Manuscript X-C-ST@1 · X-C-ST.tex · definition · b679954e

Definition: Completed test and field

The completed relational test · lines 1147–1172

X-C-ST@1#def:completed, X-C-ST@1#eq:completed, X-C-ST@1#eq:completed-recursion

Set

e [q] e_{1} [h] e_{2} [h, k] = R_{q} c [q] = f + κ e_{1} [h] + κ^{2} e_{2} [h, k], = c_{1} [h] - L_{1, g} [h] E_{A} f, = c_{2} [h, k] - L_{1, g} [h] E_{A} c_{1} [h] - L_{2, g} [h, k] E_{A} f + L_{1, g} [h] E_{A} L_{1, g} [h] E_{A} f .

Equivalently,

e_{2} = c_{2} - L_{1, g} [h] E_{A} e_{1} - L_{2, g} [h, k] E_{A} f .

The completed field is

A [q] = ϕ (e [q]) = ϕ (f) + κ A_{1} [h] + κ^{2} A_{2} [h, k] .

Displayed formulas in this passage

X-C-ST@1#eq:completed · line 1150
$e [q] e_{1} [h] e_{2} [h, k] = R_{q} c [q] = f + κ e_{1} [h] + κ^{2} e_{2} [h, k], = c_{1} [h] - L_{1, g} [h] E_{A} f, = c_{2} [h, k] - L_{1, g} [h] E_{A} c_{1} [h] - L_{2, g} [h, k] E_{A} f + L_{1, g} [h] E_{A} L_{1, g} [h] E_{A} f .$
X-C-ST@1#eq:completed-recursion · line 1162
$e_{2} = c_{2} - L_{1, g} [h] E_{A} e_{1} - L_{2, g} [h, k] E_{A} f .$
unlabeled · line 1167
$A [q] = ϕ (e [q]) = ϕ (f) + κ A_{1} [h] + κ^{2} A_{2} [h, k] .$