Discussion Overview
The discussion revolves around identifying all irreducible polynomials of degrees 1 to 5 over the finite field F2. Participants explore various solutions and methods for determining irreducibility, as well as addressing discrepancies with a textbook solution.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose that the polynomial ##X^5+X^4+X^2+X+1## is a valid degree 5 irreducible polynomial, suggesting it may have been omitted from the textbook's solution.
- Others agree with the inclusion of ##X^5+X^4+X^2+X+1## and also mention additional degree 5 polynomials: ##X^5+X^4+X^3+X+1##, ##X^5+X^3+X^2+X+1##, and ##X^5+X^4+X^3+X^2+1##.
- One participant discusses the criteria for irreducibility in polynomials over Z/2, noting that irreducible cubics must end in 1 and have an odd number of terms, providing examples such as ##X^3+X^2+1## and ##X^3+X+1##.
- There is mention of the tedious nature of checking irreducibility and references to broader challenges in algebraic geometry related to irreducibility.
Areas of Agreement / Disagreement
Participants express differing views on the completeness of the textbook's solution, with some asserting that additional irreducible polynomials exist while others seem to agree on the identified polynomials. The discussion remains unresolved regarding the totality of irreducible polynomials of degree 5.
Contextual Notes
Participants note that checking irreducibility can be inherently difficult, and there are references to unresolved mathematical steps and criteria for irreducibility that may depend on specific definitions.
