Hi rush,
Let
then the right ideal distributing will give us
Is this in the ideal? note each of the coefficients so all we need is for the coefficients to sum to 0
this places
The other half I leave up to you
Hello,
I can't seem to figure out a part of this question:
Let F be a field and let I = { a(sub n)x^n + a(sub n-1)x^(n-1) + ... + a (sub 0), where the a(sub i) are elements of F and a(sub n) + ... + a (sub 0) = 0}. Show that I is an ideal of F[x].
Here's what I have so far: So I defined an f(x) and g(x) that were elements of I and showed that f(x) - g(x) is in I. I'm trying to use the ideal test so I need to show now that, for an r(x) that's an element of F[x], r(x)f(x) and f(x)r(x) are in I. I can't figure out how to do this.
Any help is much appreciated, thanks!
Sorry i misunderstood the definition of your F[x]. Now i understand that it is the set of linear combinations of x with elements in the field F. Then,
regardless the element will be of the form
but applying the same technique as before will be a little messy so an alternative you can try is to use the fact that
Since you are just summing the coefficients of f(x). Also note this is a characteristic of all elements in the ideal I.
Using this result we can multiply the two functions and evaluate their product at 1
Hope this helps