Yes, by "the" function, I meant the natural function mapping each polynomial in F[x] to it's associated function in F^F.
For the first case, what you mentioned is true. I had figured that much myself, but that proves that F is infinite => ker E = 0 <=> E is 1-to-1. How do we go in the opposite...
Hi,
Can someone explain why the following is true? It seems to be an "accepted fact" everywhere I search, and I can't tell why.
Let F be a field. Let E be the function from F[x] to F^F, where F[x] is the set of all polynomials over F, and F^F is the set of all functions from F to F.
Then...
I follow the first 2 statements and the last statement. But I'm not sure where to apply the division algorithm.
Let me ask this...is the following the right way to go about it?
Begin with the idea of what a monic polynomial generator is. Call it d.
By the definition of I, d is in I, and so...
Yeah, that's the impression I got, but I think I might just be missing out some basic connection. So far, I have the following.
Let I be the ideal generated by f and g.
Since d = gcd(f, g), we know that d|f and d|g so d|(af+bg) for some a,b in F. And therefore d is in I.
I cannot think of...
Hi.
Given a field F and the space of polynomials over it, F[x], I was wondering why it is that the monic generator of the ideal generated by any two polynomial f, g from F[x] is exactly the gcd of f and g.
Thank you!
Kriti