AV-Portal 3.23.2 (82e6d442014116effb30fa56eb6dcabdede8ee7f)

Certifying Irreducibility in Q[x]

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Certifying Irreducibility in Q[x]
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We consider the question of certifying that a polynomial in Z[x] or Q[x] is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv. that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in F p [x] are self-certifying.
Keywords polynomial irreducibility certificate
Prime ideal Field extension Maxima and minima Mathematical analysis Divisor
Inclusion map Connected space Function (mathematics) Line (geometry) Algebraic number Divisor Category of being Factorization Abelsche Erweiterung
Divisor Sequence
Polygon Subset Prime ideal Field extension Primality test Lemma (mathematics) Resultant Divisor Mathematical analysis Content (media) p-adische Zahl Einheitswurzel
Random number Primality test Lemma (mathematics) Mathematical analysis Divisor (algebraic geometry) Product (business) Root Prime ideal Field extension Radical (chemistry) Faktorenanalyse Set theory Divisor Ranking Matrix (mathematics)
Polynomial Field extension Lemma (mathematics) Directed set Faktorenanalyse Divisor Divisor (algebraic geometry) Mathematical analysis Transformation (genetics) Determinant Singuläres Integral
Root Divisor (algebraic geometry) Divisor Transformation (genetics) Prime number
Prime ideal Radical (chemistry) Mathematical analysis Divisor Factorization Statistical hypothesis testing Singuläres Integral