# Question 33651: Frobenius Fixes Every Element of This Prime Field [Return to catalogue page 169](https://ocme.evemisslab.com/en/questions/page/169/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r06-finite-field-foundations/#q-20261014-advance-20261031-r06-finite-field-0076) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0066-33651 · q-20261014-advance-20261031-r06-finite-field-0076 ## Original question and givens Use the complete prime field Z_2, with prime p=2, 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. element=0, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p. - G001 — Use the complete prime field Z_2, with prime p=2, 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. element=0, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-finite-field](https://ocme.evemisslab.com/en/mko/mko-finite-field/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod31-fielddeep-frobenius, [domain, element, operation, p]] After derived = [fielddeep-frobenius, 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 dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001 S002 · impliesBefore source roles = [2, []] After derived = [[], 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 dependenciesRule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001 S003 · calculationBefore Field ORIGINAL previous power times element k=1 := mul(1, 0) After Field ORIGINAL previous power times element k=1 = 0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [power exponent1, 0, 2] After derived = [0, 0] Conditions: none Use integer Euclidean division: value=n*quotient+remainder with0≤remainder