Question

Write down all the monic irreducible polynomials of degree 1 in (Z/5Z)[x]. What is the product...

Write down all the monic irreducible polynomials of degree 1 in (Z/5Z)[x]. What is the product of these 5 polynomials? Do the same for degree 2 in (Z/5Z)[x].{Hint: there are 10of them} what is the product of these 10 polynomials

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
8. List all irreducible polynomials with binary coefficients of degree 4 or less. (Hint: produce a...
8. List all irreducible polynomials with binary coefficients of degree 4 or less. (Hint: produce a times table that shows the minimum number of products needed.) Show these as binary numbers (omitting the indeterminant) and as decimal numbers (interpreting the binary number into decimal). Is 23 a prime polynomial in this field? 9. Interpreting these decimal numbers into coefficients of polynomials with binary coefficients, what is the product of 11 and 10 modulo 31 in GF(2^4) over P = 31?...
Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials...
Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials in Z5[x], i.e. represent f(x) as a product of irreducible polynomials in Z5[x]. Demonstrate that the polynomials you obtained are irreducible. I think i manged to factorise this polynomial. I found a factor to be 1 so i divided the polynomial by (x-1) as its a linear factor. So i get the form (x3 − 2x2 + 2x − 1) = (x2-x+1)*(x-1) which is...
1)T F: All (x, y, z) ∈ R 3 with x = y + z is...
1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3 9 2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R 3 3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here M22 is the vector space of all 2 × 2 matrices.) 4) T F: All polynomials of degree exactly 3 is...
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that...
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that p(-1)=0. (a) Prove that S is a subspace of the vector space of all polynomials. (b) Find a basis for S. (c) What is the dimension of S? 6. Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2 =(1,2,-6,1), ?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2 =(3,1,2,-2). Prove that V=W.
Solve each system by elimination. 1) -x-5y-5z=2 4x-5y+4z=19 x+5y-z=-20 2) -4x-5y-z=18 -2x-5y-2z=12 -2x+5y+2z=4 3) -x-5y+z=17 -5x-5y+5z=5...
Solve each system by elimination. 1) -x-5y-5z=2 4x-5y+4z=19 x+5y-z=-20 2) -4x-5y-z=18 -2x-5y-2z=12 -2x+5y+2z=4 3) -x-5y+z=17 -5x-5y+5z=5 2x+5y-3z=-10 4) 4x+4y+z=24 2x-4y+z=0 5x-4y-5z=12 5) 4r-4s+4t=-4 4r+s-2t=5 -3r-3s-4t=-16 6) x-6y+4z=-12 x+y-4z=12 2x+2y+5z=-15
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials...
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials in each of the following rings, justifying your answers briefly: (i) Z3 [x]; (ii) Q[x] (this can be done easily using an appropriate theorem); (iii) R[x] (hints: you may find it helpful to write γ = 2^(1/4), the positive real fourth root of 2, and to consider factors of the form x^2 + a*x + 2^(1/2); (iv) C[x] (you may leave your answer in...
Find a basis for the set of all third degree polynomials which contains the set {x...
Find a basis for the set of all third degree polynomials which contains the set {x + 1, x2 - x + 1}.
Consider interval [0,2] and an inner product with weight w(x)=1. Starting with 1, x, x^2, x^3...
Consider interval [0,2] and an inner product with weight w(x)=1. Starting with 1, x, x^2, x^3 and using Gram-Schmidt process, build four polynomials of degree 0,1,2,3 orthogonal on [0,2]. You do not have to normalize them (I think this simplifies calculations). Just remember that if they are not normalized, Gram-Schmidt formula will have denominators in the form <q_j,q_j>. (If you do normalize them, then <q_j,q_j> = 1).
If p(x) and q(x) are arbitrary polynomials of degree at most 2, then the mapping =p(−1)q(−1)+p(0)q(0)+p(2)q(2)...
If p(x) and q(x) are arbitrary polynomials of degree at most 2, then the mapping =p(−1)q(−1)+p(0)q(0)+p(2)q(2) defines an inner product in P3. Use this inner product to find , ||p||, ||q||, and the angle θ between p(x) and q(x) for p(x)=2x^2+3 and q(x)=2x^2−6x.
Let P2 denote the vector space of polynomials in x with real coefficients having degree at...
Let P2 denote the vector space of polynomials in x with real coefficients having degree at most 2. Let W be a subspace of P2 given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper subspace of P2.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT