Are Algebraic Extensions Always Finite?

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The discussion centers on the nature of algebraic extensions and their properties. It clarifies that not every algebraic extension is finite, with the algebraic closure of the rationals serving as a counterexample. The conversation also addresses the concept of algebraically closed fields, noting that a field is algebraically closed if every nonconstant polynomial in its polynomial ring factors into linear factors. Participants explore the implications of field characteristics, particularly in relation to finite fields and their algebraic closures. The discussion concludes with a deeper inquiry into the characteristics of fields containing finite fields, emphasizing the relationship between a field's structure and its algebraic properties.
  • #31
Hurkyl said:
How? Can you come up with an example?

The dot notation is definitely not always compatible with field notation, as I will show. Consider the field Z/3Z. Writing 5*2 is absurd because 5 is not an element of the field Z/3Z. You need to write 5 \cdot 2 if you want to represent the sum 2+2+2+2+2.

morphism, if you are saying that if F is a field that contains f, then n \cdot f = nf WHEN the characteristic of F is greater than n or equal to 0, then I think that is true.
 
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  • #32
ehrenfest said:
Writing 5*2 is absurd because 5 is not an element of the field Z/3Z.[/itex]
Er, yes it is. 5 denotes the same element that 2 does.
 
  • #33
OK. This is getting really technical. I would rather move on.
 

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