let f(x) and g(x) be two polynomials with rational coefficients. Let d(x) be the greatest common of f(x) and g(x) in Q[x] (Q as in the set of rational numbers) and e(x) the greatest common divisor of f(x) and g(x) in C[x] (C and in set of complex numbers). is d(x) = e(x)
Let f(x) and g(x) be two polynomials with rational coefficients. Let d(x) be the greatest common ...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
Which of the following polynomials is a greatest common divisor of the two elements x3-1 and x2-1 of the ring Q[x]? x+1 x-1 o 1 x2+1
88. Let D be an integral domain. (a) For a, b E D define a greatest common divisor of a and b. (b) For rE D denote (x)dr dE D.Prove that if (a) +(b)- (d), then d is a greatest common divisor of a and b. 88. Let D be an integral domain. (a) For a, b E D define a greatest common divisor of a and b. (b) For rE D denote (x)dr dE D.Prove that if (a) +(b)-...
1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K] < oo. 1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K]
Let (a, b) denote the greatest common divisor (ged) of the numbers a and b. Let x ((61610+1,6171-1)61 +1, (61611,61671)610- 610 1,61 (a) Find X mod 10 (b) What is the minimum number of bits required to represent X? Let (a, b) denote the greatest common divisor (ged) of the numbers a and b. Let x ((61610+1,6171-1)61 +1, (61611,61671)610- 610 1,61 (a) Find X mod 10 (b) What is the minimum number of bits required to represent X?
Let (a, b) denote the greatest common divisor (ged) of the numbers a and b. Let x ((61610+1,6171-1)61 +1, (61611,61671)610- 610 1,61 (a) Find X mod 10 (b) What is the minimum number of bits required to represent X? Let (a, b) denote the greatest common divisor (ged) of the numbers a and b. Let x ((61610+1,6171-1)61 +1, (61611,61671)610- 610 1,61 (a) Find X mod 10 (b) What is the minimum number of bits required to represent X?
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
Let f(x) = ax^2 +bx +c be a quadratic whose coefficients a, b, c are rational. Prove that if f(x) has one rational root, then the other root is also rational.
2. Let P3 stand for the vector space of all polynomials in x with real coefficients and of the degree at most 3. (a) (1 mark) Show that the set E = {p(x) € P3 : p(3)=0}, is a subspace of P3. (b) (2 marks) Show that the collection of polynomials {(x - 3), (x – 3), (x-3)3} is a basis of E.
6. Let Pm (F) be the vector space of polynomials p(x) = ao + a1x + ... Amx" with coefficients in F and degree at most m, and let U be the set of even polynomials in P5(F): U := {p(x) € P5(F) | P(x) = p(-x)}. (a) Show that the list of vectors 1, x, x², x3, x4 + x, x + x spans P5(F). (b) Show that U is a vector subspace of P5(F) (c) Prove that there...