Certifying Irreducibility in Q[x]
Formal Metadata
Title 
Certifying Irreducibility in Q[x]

Title of Series  
Author 

License 
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 

Release Date 
2020

Language 
English

Production Year 
2020

Content Metadata
Subject Area  
Abstract 
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 selfcertifying.

Keywords  polynomial irreducibility certificate 
00:00
Prime ideal
Field extension
Maxima and minima
Mathematical analysis
Divisor
00:25
Inclusion map
Connected space
Function (mathematics)
Line (geometry)
Algebraic number
Divisor
Category of being
Factorization
Abelsche Erweiterung
02:57
Divisor
Sequence
06:02
Polygon
Subset
Prime ideal
Field extension
Primality test
Lemma (mathematics)
Resultant
Divisor
Mathematical analysis
Content (media)
padische Zahl
Einheitswurzel
09:56
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)
13:34
Polynomial
Field extension
Lemma (mathematics)
Directed set
Faktorenanalyse
Divisor
Divisor (algebraic geometry)
Mathematical analysis
Transformation (genetics)
Determinant
Singuläres Integral
16:09
Root
Divisor (algebraic geometry)
Divisor
Transformation (genetics)
Prime number
18:33
Prime ideal
Radical (chemistry)
Mathematical analysis
Divisor
Factorization
Statistical hypothesis testing
Singuläres Integral