Original question and givens
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=13, b=13, c=25; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
G001 — Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=13, b=13, c=25; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod31-fielddeep-affine-solve, [a, b, c, domain, operation, p]]
After
derived = [fielddeep-affine-solve, 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 = [41, [2, 3, 4, 5, 6]]
After
derived = [[1, 2, 1, 1, 5], 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 · calculation
Before
Field ORIGINAL a times proposed inverse := mul(13, 19)
After
Field ORIGINAL a times proposed inverse = 247
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [inverse substitution, 247, 41]
After
derived = [6, 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 · calculation
Before
Field ORIGINAL c minus b := sub(25, 13)
After
Field ORIGINAL c minus b = 12
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Field ORIGINAL inverse times c minus b := mul(19, 12)
After
Field ORIGINAL inverse times c minus b = 228
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
source roles = [affine solution, 228, 41]
After
derived = [5, 23]
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 · calculation
Before
Field ORIGINAL a times solution := mul(13, 23)
After
Field ORIGINAL a times solution = 299
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Field ORIGINAL a solution plus b := sum(299, 13)
After
Field ORIGINAL a solution plus b = 312
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [equation substitution, 312, 41]
After
derived = [7, 25]
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 = [41, 13, 13, 25, 19]
After
derived = [23, 25, Yes]
Conditions: none
Prime p and nonzero a give an inverse.Subtract b,then multiply by it and substitute under the SAME mod-p operations.If x,z solve,then a(x−z)=0;inverse multiplication gives x=z,proving uniqueness.Never divide by0.
Step identifiers and dependencies
Rule: Field_INVERT_nonzero_coefficient_and_UNIQUENESS · Uses: G001, S010