Determination of irreducible polynomials over a given field

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SUMMARY

This discussion focuses on determining irreducible polynomials of the form x² + ax + b over the finite field F3. The user presents specific examples, including x² + x + 2, which is confirmed as irreducible, and seeks clarification on the irreducibility of x² + x + 1 over F2. The key conclusion is that a polynomial is reducible if it has a root in the field, specifically if one of the elements {0, 1, 2} is a solution for the quadratic equation.

PREREQUISITES
  • Understanding of finite fields, specifically F3 and F2
  • Knowledge of polynomial factorization techniques
  • Familiarity with the concept of irreducibility in polynomial algebra
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of finite fields, particularly F3 and F2
  • Learn about polynomial factorization in finite fields
  • Explore the criteria for irreducibility of polynomials over finite fields
  • Investigate the application of the Rational Root Theorem in polynomial equations
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in polynomial theory and finite field applications.

catcherintherye
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I am required to find all irreducible polynomials of the form xsquared + ax + b over the field F3, I have the 9 cases infront of me, i can see when something is reducible say xsquared is p(x)q(x) where p=x, q=x, but i have particular difficulty seeing when something is irreducible, e.g i know that xsqd + x + 2 is but i don't know how to show it, just as i do not know how xsqd + x +1 is irreducible over F2, although i can see how xsqd + 1 is since xsqd +1 = xsqd + 2x + 1 =(x+1)sqd, how do i know that a similar trick could not have been employed to make xsqd + x + 1 factorise?
 
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It's just a quadratic. If it is reducible then the factors are linear. So it is reducible if and only if one of 0,1,2 is a solution.
 

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