How are multiplication tables for fields created?

[phillyj]

The class I'm in is not modern algebra nor have I take those courses. The professor of the class [Learning how to read/write in math] decided to try to teach some abstract algebra. I am trying to understand how multiplication is done in a field. I am using this site to help me as it was most relevant to making tables like the professor wanted:

http://math.arizona.edu/~ura-reports/041/Patterson.Genevieve/Final/FinalReport/node8.html

I am stuck on 9 element fields. This is the second to last example at the end of the page. I understand that we can extend the F[3] field to get the table.

The site says "As a vector space, F[9] = F[3^2] = {a+bx, (a,b)\in F[3]}. To find the multiplication table we need a monic-quadratic that has no zeros in F[3]
A monic-quadratic will have a coefficient of 1 on the highest degree term."

[1]Why do they use monic quadratic?

[2] When they try x^2+1, they get f(0)=1, f(1)=2 but why is f(2)=2? Are they not plugging into x?

I don't really have good study material as the professor wrote a short paper on this stuff but it is bare minimum.

Thanks for your help.

 

[Office_Shredder]

F[3] is just the integers modulo 3. Given a number, its remainder when you divide by 3 is either 0, 1 or 2, and that's the corresponding number that you get in F[3]. So f(2)=5, but we have to take the remainder when you divide by 3, which gives f(2)=2.

The choice for a monic quadratic is just that it makes the calculations a little easier to look at (there are fewer coefficients flying around). x is going to be an element that satisfies f(x)=0. If we have ax^2+bx+c=0 the way we use this is by re-arranging:
x^2 = \frac{1}{a}(-bx-c)

So given any arbitrary polynomial in x, we can write it as something of the form cx+d by repeatedly replacing x²s with the other side of that equality. We could have chosen a monic f(x) by just dividing both sides of ax^2+bx+c=0 and we would have ended up with the same solution for x² at the end anyway. So we can basically just ignore the a if we assume our polynomial is monic to begin with