Purchase Solution

Polynomial Rings: Irreducible Polynomials

Not what you're looking for?

Ask Custom Question

Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.

Attachments
Purchase this Solution
Solution Summary

Irreducible polynomials are investigated in this solution, which is detailed and well presented.

Solution Preview

Modern Algebra
Ring Theory (XXIX)
Polynomial Ring
Irreducible Polynomial

By:- Thokchom Sarojkumar Sinha

Prove that is irreducible over the field of integers mod
and prove directly that is a field having 121 elements.

Solution:- Here .
The field is .
Let , the ring of polynomials having coefficients in .
is a factor of if and only if for some .

Here
Hence

Therefore

...

Solution provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
Recent Feedback
  • "Thanks this really helped."
  • "Sorry for the delay, I was unable to be online during the holiday. The post is very helpful."
  • "Very nice thank you"
  • "Thank you a million!!! Would happen to understand any of the other tensor problems i have posted???"
  • "You are awesome. Thank you"
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Probability Quiz

Some questions on probability

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Polynomial rings

    Hence is irreducible in . (b) For , we set , then is equal to in (a). Since is irreducible in , then is irreducible in . Done. This shows how to prove that given polynomials are irreducible.

  • Irreducible Polynomials

    122314 Irreducible Polynomials I need to find the irreducible polynomial in Z3[x].

  • Irreducible Palindromic Polynomials, Reducible and Irreducible Polynomials and Degrees of Polynomials

    From problem #3(c), is irreducible. Irreducible Palindromic Polynomials, Reducible and Irreducible Polynomials and Degrees of Polynomials are investigated. The solution is detailed and well presented.

  • Irreducible polynomial problems

    471901 Irreducible polynomial problems Determine the number of monic irreducible polynomials of degree 4 in F_q without using the Moebius Inversion Formula. Let us first calculate the number of reducible monic polynomials of degree 4 over F_q.

  • Irreducible polynomials

    397860 Discussing irreducible polynomials a. Let F be a field and f (x) an irreducible polynomial of degree 3 in F [x]. Show that if K is an extension of F of dimension 10, then f(x) is irreducible in K[x]. b.

  • Primitive Irreducible Polynomials

    106282 Primitive Irreducible Polynomials Please can you explain primitive irreducible polynomials and please give examples. Please see the attached file for the fully formatted problem. Please see the attached file for the complete solution.

  • Irreducible Polynomials

    53606 Irreducible Polynomials Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.

  • Irreducible Polynomials

    Find, with justification, a polynomial in that is irreducible over but not over . Please see the attached file for the complete solution. 1 Decide, giving your reasons, which of the following polynomials is irreducible over .

  • Irreducuble Polynomials : Find the Minimum Polynomial and Factorization

    58986 Irreducible Polynomials : Minimum Polynomial and Factorization Find (with proof) an element a Є Q such that Q(√3, √10)= Q(a) . Find the minimum polynomial of √3 + √10 . Hence, or otherwise, show that x^4 -26x^2 + 49 is irreducible over Q.

View More Related Solutions