13-04 Splitting Fields and Algebraic Closures

Definition. algebraic closure

• $F$,
  • field

$\bar{F}$ is said to be algebraic clusre over $F$ if

• (1) $\bar{F}$ is algebraic over $F$,
• (2) For every $f \in F[x]$, $f$ splits completely over $\bar{F}$,

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Definition. algebraically closed

• $F$,
  • field

$F$ is said to be algebraically closed if for every $f \in F$ the all roots of $f$ are in $F$.

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