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Dummit and Foote in their book Abstract Algebra give the following definition of an algebraically closed field ... ...

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

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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