Dummit and Foote in their book Abstract Algebra give the following definition of an algebraically closed field ... ...(adsbygoogle = window.adsbygoogle || []).push({});

From the remarks following the definition it appears that the definition only applies to ##K[x]## ...

Does it also apply to ##K[x_1, x_2], K[x_1, x_2, x_3], \ ... \ ... \ , K[x_1, x_2, \ ... \ ... \ , x_n] , \ ... \ ... \ ...## ?

That is ... when we say K is an algebraically closed field does it imply that every polynomial in ##K[x_1, x_2, \ ... \ ... \ , x_n]## has a root in ##K## ... ... ?

... ... or maybe it is better if I say ... how does the definition of algebraically closed generalise to ##K[x_1, x_2, \ ... \ ... \ , x_n]## ... ... ?

Hope someone can clarify this issue ... ...

Peter

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# I Algebraically Closed Fields

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