Question

In: Advanced Math

A First Course in Abstract Algebra Chapter S.22, Problem 27E Let F be a field of...

A First Course in Abstract Algebra Chapter S.22, Problem 27E

Let F be a field of characteristic zero and let D be the formal polynomial differentiation map, so that. D(a0 + a1x + a2x^2 + ••• + anx^n) = a1 + 2 • a2x + •••+n• anxn-1, i.e.

F be a field of characteristic zero , D:F[x]→F[x].

C) Find the image of F[x] under D. Is it important which characteristic the field has? Can you explain this enough?

http://www.chegg.com/homework-help/wooden-cubes-size-painted-different-color-face-make-children-chapter-S.22-problem-27E-solution-9780201763904-exc

Expert Solution

is the polynomial differentiation map

Then, the image of under this map is given by the set of polynomials

If is a polynomial then

Therefore, the image consists of polynomials of the form

If the field has characteristic 2 for example, then the polynomial having the component because characteristic 2 means in this field (definition of characteristic)

Thus, the image reduces to polynomials of the form if the characteristic is 2 instead of 0. We can see that the characteristic does play a role in defining the image of our map

Please don't forget to rate positively if you found this response helpful. Feel free to comment on the answer if some part is not clear or you would like to be elaborated upon. Thanks and have a good day!

Colby Messinger answered 2 years ago

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the...

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold: (a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x). (b) If g(x)|f(x), then g(x)h(x)|f(x)h(x). (c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x). (d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS. (ii)...

Abstract Algebra Let G be a discrete group of isometries of R2. Prove there is a...

Abstract Algebra Let G be a discrete group of isometries of R2. Prove there is a point p ∈ R2 whose stabilizer is trivial.

Let F be a field and let φ : F → F be a ring isomorphism....

Let F be a field and let φ : F → F be a ring isomorphism. Define Fix φ to be Fix φ = {a ∈ F | φ(a) = a}. That is, Fix φ is the set of all elements of F that are fixed under φ. Prove that Fix φ is a field. (b) Define φ : C → C by φ(a + bi) = a − bi. Take for granted that φ is a ring isomorphism (we...

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following...

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following sets. (a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)} (b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is...

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is a subgroup of G1. Define φ(H) = {φ(h) | h ∈ H}. Prove that φ(H) is a subgroup of G2. (b) Let ker(φ) = {g ∈ G1 | φ(g) = e2}. Prove that ker(φ) is a subgroup of G1. (c) Prove that φ is a group isomorphism if and only if ker(φ) = {e1} and φ(G1) = G2.

Question

A First Course in Abstract Algebra Chapter S.22, Problem 27E Let F be a field of...

Solutions

Expert Solution

Related Solutions