Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod31-fielddeep-characteristic, [domain, operation, p]]
After
derived = [fielddeep-characteristic, Exact_prime_field_Zp]
Conditions: none
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Step identifiers and dependencies
Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001
S002 · implies
Before
source roles = [11, [2, 3]]
After
derived = [[1, 2], Yes]
Conditions: none
Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field.
Step identifiers and dependencies
Rule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001
S003 · implies
Before
source roles = [successive sum of one k=0, 0, 11]
After
derived = [0, 0]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S002
S004 · implies
Before
source roles = [successive sum of one k=1, 1, 11]
After
derived = [0, 1]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003
S005 · implies
Before
source roles = [successive sum of one k=2, 2, 11]
After
derived = [0, 2]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S004
S006 · implies
Before
source roles = [successive sum of one k=3, 3, 11]
After
derived = [0, 3]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005
S007 · implies
Before
source roles = [successive sum of one k=4, 4, 11]
After
derived = [0, 4]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S006
S008 · implies
Before
source roles = [successive sum of one k=5, 5, 11]
After
derived = [0, 5]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007
S009 · implies
Before
source roles = [successive sum of one k=6, 6, 11]
After
derived = [0, 6]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S008
S010 · implies
Before
source roles = [successive sum of one k=7, 7, 11]
After
derived = [0, 7]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009
S011 · implies
Before
source roles = [successive sum of one k=8, 8, 11]
After
derived = [0, 8]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S010
S012 · implies
Before
source roles = [successive sum of one k=9, 9, 11]
After
derived = [0, 9]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011
S013 · implies
Before
source roles = [successive sum of one k=10, 10, 11]
After
derived = [0, 10]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S012
S014 · implies
Before
source roles = [successive sum of one k=11, 11, 11]
After
derived = [1, 0]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013
S015 · implies
Before
source roles = [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0]]
After
derived = [11]
Conditions: none
List sums of0 through p copies of1.Residues1..p−1 are nonzero and p returns0,so the FIRST POSITIVE additive return is p.Zero is not a positive return;this is not multiplicative order.
Step identifiers and dependencies
Rule: Field_CHARACTERISTIC_first_POSITIVE_additive_return · Uses: G001, S014
Result
(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.