Let $F$ be a field.

I already know that there is an algebraic closure $\overline{F}$ of $F$ which satisfy that every $f(x)\in \overline{F}[x]$ is splits in $\overline{F}[x]$.

Is there a smallest extension field $E$ of $F$ such that every $f(x)\in {\mathbf F}[x]$ is splits in $E[x]$?
Or such extension field $E$ must be the algebraic closure of $F$?

  • 2
    $\begingroup$ The algebraic closure is the smallest such extension by definition. $\endgroup$ – Alex G. Sep 26 '15 at 15:25
  • $\begingroup$ Which raises the question; what is your definition of an algebraic closure? $\endgroup$ – Servaes Sep 26 '15 at 15:31
  • $\begingroup$ @AlexG. The ``smallest'' means that every $f(x)\in \overline{F}[x]$ is splits in $\overline{F}[x]$. But I only want to every $f(x)\in {\mathbf F}[x]$ is splits in $E[x]$. $\endgroup$ – bfhaha Sep 26 '15 at 15:34
  • $\begingroup$ @Servaes The algebraic closure of $F$ is an algebraic extension of $F$ which is algebraically closed. $\endgroup$ – bfhaha Sep 26 '15 at 15:41

Let $E$ be the smallest extension of $F$, where every polynomial in $F[X]$ splits. Then $E$ is algebraically closed, i.e., every polynomial in $E[X]$ splits as well: Consider $f\in E[X]$. Its finitely many coefficients are in a finite algebraic extension $K$ of $F$ (presumably smaller than $E$). If we multiply together all the (finitely many) conjugates of $f$, we obtain a polynomial that is in fact in $F[X]$, so splits by assumption; consequently its factor $f$ splits as well.

This argument uses Galois theory; if you don't have enough of it available, you may have to gnaw yourself through the base of $K/F$ step by step ...

  • $\begingroup$ Thank you. Which book can I find the detail of this argument. Like the definition of the conjugates of $f$ and the base of $K/F$, etc. $\endgroup$ – bfhaha Sep 26 '15 at 15:57
  • $\begingroup$ If you are learning field-theory and field-extensions from any book, it should be in there, I guess $\endgroup$ – Hagen von Eitzen Sep 27 '15 at 8:19
  • $\begingroup$ I find out the conjugate of an element (another root of its minimal polynomial), and the conjugate of an algebraic extension (its image under a automorphism). But I can't find any definition about the conjugate of a polynomial $f$. Could you please give me a definite book name or is there another name of the conjugate of a polynomial. Thank you very much. $\endgroup$ – bfhaha Sep 30 '15 at 10:17

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