# Homework Help: Construct fields of each of the following orders

1. Nov 4, 2008

### mathbbb2

Construct fields of each of the following orders: $$\textbf{(a)}$$ 9 $$\textbf{(b)}$$ 49 $$\textbf{(c)}$$ 8 $$\textbf{(d)}$$ 81 (you may exhibit these as $$\frac{F[x]}{(f(x))}$$ for some $$F$$ and $$f$$).

Relevant Theorems to use:
$$\textbf{(1.)}$$ Let $$f(x)$$ be a polynomial in $$F[x]$$. $$\frac{F[x]}{(f(x))}$$ is a field iff $$f(x)$$ is irreducible.

$$\textbf{(2.)}$$ $$F$$ is a finite field of order $$q$$ and let $$f(x)$$ be a polynomial in $$F[x]$$ of degree $$n \geq 1$$. Then $$\frac{F[x]}{(f(x))}$$ has $$q^n$$ elements
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 4, 2008

### Office_Shredder

Staff Emeritus
I think you should use the two theorems