Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$?
Note: I am not looking for the algebraic closure of $F$.
One candidate is the field of rational functions over $F$, but I have a problem in my argument when the numerator and denominator are the same degree.
Does anyone have a more elegant Argument than multiplying out powers of rational functions?
2026-04-02 06:58:25.1775113105
Proper Field extensions
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Being a root of a polynomial in $F[x]$ is called being algebraic over $F$. There are extensions $K/F$ in which no element of $K\setminus F$ is algebraic over $F$. Indeed any purely transcendental extension will work (i.e. any adjunction of a collection of algebraically independent indeterminates - no matter what the cardinality of this collection). This is because any rational function satisfying an algebraic relation over $F$ would, after clearing denominators, exhibit an algebraic relation among the hypothesized algebraic independents, a contradiction.