Thread: Rings and fields - Write down the nine elements of F9[i] help

1. Rings and fields - Write down the nine elements of F9[i] help

In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
the set F9[i] consisting of all numbers a+bi, with a,b in F9. Add and multiply these
numbers as though they were polynomials in 'i', except whenever you get i^2 replace
it by −1.
(i) Write down the nine elements of F9[i] .
(ii) Show that every nonzero element of F9[i] has an inverse, so that F9[i] is a
field.

I know im supposed to show you that ive tried the question if i want an answer. Believe me, i have tried it. Im just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, im pretty sure id be able to get part b on my own.

Thanks

2. Originally Posted by habsfan31
In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
the set F9[i] consisting of all numbers a+bi, with a,b in F9. Add and multiply these
numbers as though they were polynomials in 'i', except whenever you get i^2 replace
it by −1.
(i) Write down the nine elements of F9[i] .
(ii) Show that every nonzero element of F9[i] has an inverse, so that F9[i] is a
field.

I know im supposed to show you that ive tried the question if i want an answer. Believe me, i have tried it. Im just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, im pretty sure id be able to get part b on my own.

Thanks

Say $\mathbb{F}_3:\{0,1,2\}$ , with operations modulo 3, so $F_3[i]:=\{a+bi / a,\,b\in \mathbb{F}_3\}$ , with operations

modulo 3 and also taking into account that $i^2=-1=2\!\!\pmod 3$ .

1) Show first that indeed $|\mathbb{F}_3[i]|=9$ ;

2) Show that all the axioms of field are fulfilled in $\mathbb{F}_3[i]$ , in particular identify the neutral

elements wrt addition and multiplication (it's basically the same as with the "usual" complex numbers)

3) Now show that every non-zero element has an inverse (again, $\mathbb{C}$ is a good example)

Tonio