Original question and givens
Use the complete prime field Z_73, with prime p=73, 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=21, b=21, c=41; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).
G001 — Use the complete prime field Z_73, with prime p=73, 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=21, b=21, c=41; 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 = [73, [2, 3, 4, 5, 6, 7, 8]]
After
derived = [[1, 1, 1, 3, 1, 3, 1], 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(21, 7)
After
Field ORIGINAL a times proposed inverse = 147
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, 147, 73]
After
derived = [2, 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(41, 21)
After
Field ORIGINAL c minus b = 20
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(7, 20)
After
Field ORIGINAL inverse times c minus b = 140
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, 140, 73]
After
derived = [1, 67]
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(21, 67)
After
Field ORIGINAL a times solution = 1407
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(1407, 21)
After
Field ORIGINAL a solution plus b = 1428
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, 1428, 73]
After
derived = [19, 41]
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 = [73, 21, 21, 41, 7]
After
derived = [67, 41, 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