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?

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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 1 year ago

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