Question

In: Advanced Math

Let Z2 [x] be the ring of all polynomials with coefficients in Z2. List the elements of the field Z2 [x]/〈x2+x+1〉, and make an addition and multiplication table for the field.

 

Let Z2 [x] be the ring of all polynomials with coefficients in Z2. List the elements of the field Z2 [x]/〈x2+x+1〉, and make an addition and multiplication table for the field. For simplicity, denote the coset f(x)+〈x2+x+1〉 by (f(x)) ̅.

Solutions

Expert Solution


Related Solutions

Write out the addition and multiplication tables for the ring Z2[x]/(x^2 + x). Is Z2[x]/(x^2 +...
Write out the addition and multiplication tables for the ring Z2[x]/(x^2 + x). Is Z2[x]/(x^2 + x) a field?
Let Z* denote the ring of integers with new addition and multiplication operations defined by a...
Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*. Can someone please explain this to me? I get that f(1) = 0, f(2) = -1 but then f(-1) = -f(1) = 0 and f(2) = -f(2) = 1 but this does not make sense in order to...
Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and...
Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and that I a subset of R is an ideal in R. In other words, suppose that I is a subring of R such that (x is in I and y is in R) implies x · y is in I. Define the relation “~” on R by y ~ x if and only if y − x is in I, and assume for the...
Let K be a field. Observe that the polynomials in K[x] that are not zero and...
Let K be a field. Observe that the polynomials in K[x] that are not zero and not units are precisely the polynomials of positive degree.
In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...
In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...
Let P denote the vector space of all polynomials with real coefficients and Pn be the...
Let P denote the vector space of all polynomials with real coefficients and Pn be the set of all polynomials in p with degree <= n. a) Show that Pn is a vector subspace of P. b) Show that {1,x,x2,...,xn} is a basis for Pn.
Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...
Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.
Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...
Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...
2 Let F be a field and let R = F[x, y] be the ring of...
2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...
1. Write down the addition and multiplication table for Z/5Z. All classes should be written in...
1. Write down the addition and multiplication table for Z/5Z. All classes should be written in terms of their canonical representative (unique representative between 0 and 4). 2. Suppose a ≡ a' mod n and b ≡ b' mod n. (a) Show that a + b ≡ a' + b' mod n. (b) Show that a · b ≡ a' · b' mod n. (An important consequence of this exercise is that addition and multiplication define maps Z/nZ × Z/nZ...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT