- #1
mathbbb2
- 1
- 0
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
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
