Listing Elements in a Finite Extension


Theorem.  Let $f \in F[X]$ be irred of degree $n$ & let $K = F[X]/(f)$.  Let $\theta = x + (f) \in K = F[X]/(f)$.  Then the elements $\{1, \theta, \theta^2, \dots, \theta^{n-1}\}$ are a basis for $K$ as a vector space over $F$ so that $[K:F] = n$ and $K = \{a_0 + a_1 \theta + \dots + a_{n-1}\theta^{n-1} : a_i \in F\}$.

Proof.  $F[X]$ is a Euclidean domain so divide $g\in F[X]$ by $f$ using the Euclidean algorithm.  The remainder is a coset representative for $g + (f)$.

Examples:
  • $\Bbb{R}[X]/(X^2 + 1) \simeq \Bbb{C}$
  • $\Bbb{Q}[X]/(X^2 + 1) \simeq \Bbb{Q}(i)$ so $[\Bbb{Q}(i) : \Bbb{Q}] = 2$

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