MATH3411 Chap.5 Number Theory with Finite Fields
Number Theory with Finite Fields
This chapter supplies the algebra behind later codes and ciphers. Euclid and Bezout identities produce greatest common divisors and modular inverses. Polynomial arithmetic over prime fields leads to irreducible and primitive polynomials, and quotienting by an irreducible polynomial creates a finite field. Primality testing and factoring complete the computational toolkit within the current examinable boundary.
Invertibility is the unifying question. An integer residue can be cancelled exactly when its gcd with the modulus is one, and a polynomial residue is invertible exactly when it is coprime to the defining polynomial. Irreducibility ensures that every nonzero class in the quotient has an inverse.
Written work should retain the Euclidean remainder chain, show back-substitution and verify the resulting coefficient by multiplication. Field computations require two visible reductions: coefficients are reduced in the base field, while polynomial powers are reduced by the chosen modulus.
Primality and factoring questions use related arithmetic but make different claims, so a compositeness witness must not be promoted to a complete factorisation or a passed base test to an unconditional proof.
What this chapter covers
- 01
Divisibility and Euclid
- 02
Bezout identities
- 03
Modular inverses
- 04
Polynomial arithmetic over prime fields
- 05
Irreducible and primitive polynomials
- 06
Finite-field construction
- 07
Primality tests and factoring
Worked example · free
Recover the Inverse of 37
- +1Run the Euclidean remainder chain to gcd one.
- +1Back-substitute to obtain 1=11(101)-30(37).
- +1Reduce -30 to 71 and verify 37(71) has remainder one.
Key terms
- Bezout identity
- An expression of a gcd as an integer linear combination.
- Irreducible polynomial
- A nonconstant polynomial with no factorisation into lower positive degrees over the base field.
- Primitive element
- A field element whose powers generate every nonzero field element.
Number Theory with Finite Fields FAQ
When may I cancel in modular arithmetic?
You may cancel a factor when it is invertible modulo the modulus, equivalently when its gcd with the modulus is one. Otherwise cancellation can discard valid residue classes or create a false conclusion.
How is a finite extension field represented?
Elements are polynomial residues of degree below a chosen irreducible polynomial, with coefficients in the base field. Addition is coefficientwise, while multiplication requires reduction by the defining irreducible polynomial.
Does passing a Fermat test prove primality?
Not in general. Some composite numbers pass selected base tests, so the conclusion depends on the precise test and assumptions. A passing result is evidence unless the stated theorem supplies a proof certificate.
Exam move
Treat integer and polynomial arithmetic as parallel algorithms with different objects. Begin an integer session by running Euclid on two fresh numbers and writing every quotient and remainder. State the gcd before asking for an inverse. If it is one, back-substitute without skipping equations, reduce the coefficient to the requested residue range and multiply the candidate by the original number.
If the gcd is larger, explain why the congruence for an inverse is impossible instead of continuing mechanically. Practise linear congruences where the gcd divides the right side as well as cases where it does not; cancellation can change the modulus and solution count, so record each condition explicitly. For finite fields, announce the base field and the irreducible reduction polynomial.
Represent every element by a polynomial below the defining degree. Perform coefficient arithmetic in the base field, then reduce powers of x using the modulus polynomial. In characteristic two, mark where addition equals subtraction, but do not let that shortcut hide the polynomial division. Build a small multiplication table only after individual reductions are reliable.
Find a polynomial inverse with Euclid and verify that the reduced product is the constant one. Exponent practice should distinguish reduction of field-element powers by the order of the nonzero multiplicative group from reduction of polynomial powers by the defining relation. For primality work, list exactly what a test proves.
Trial division uses the square-root bound; a Fermat witness proves compositeness, while passing one selected base is not a universal primality certificate. Keep factoring goals separate from primality decisions even when the same divisibility calculations appear.
Maintain an error ledger for an inverse used without a gcd check, a sign lost during back-substitution, coefficients reduced under the wrong prime, an extension polynomial assumed irreducible without justification, or a probable-prime result overstated as proof. A mixed revision block should move from integer Euclid to polynomial reduction, then to one field-power calculation and one precise test conclusion.
End each answer with a multiplication, substitution or divisibility check. This habit prepares the algebra required by BCH syndromes and RSA keys while keeping the logical status of every computation visible.
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