Section outline

  • We continued our study of algebraic extensions.

    We proved that \(K/F\) is finite if and only if it is algebraic and finitely generated.

    For a tower \(F \subseteq L \subseteq K\), we proved that \(K/F\) is algebraic if and only if \(K/L\) and \(L/F\) are algebraic. In case they are finite, we proved that \([K:F] = [K:L] \cdot [L:F]\).

    We defined the splitting field of a polynomial \(f \in F[x]\) over the field \(F\). We proved its existence. We saw examples.

    Reading: pages 5-7 from the notes. 

    Reading: pages 7-9 from the notes.