Simple Field Extension Question

In summary, the conversation discusses the proof that if u is an element of a field extension K of a field F and u^2 is algebraic over F, then u is also algebraic over K. The speaker initially considers using the contrapositive but is unsure of how to proceed. They suggest using the polynomial f(x^2) where f(u^2)=0 to prove the statement. The other speaker realizes the simplicity of this approach and thanks them for their help.
  • #1
jeffreydk
135
0
I'm currently trying to prove that (for a field extension [itex]K[/itex] of the field [itex]F[/itex]) if [itex]u\in K[/itex] and [itex]u^2[/itex] is algebraic over [itex]F[/itex] then [itex]u[/itex] is algebraic over [itex]K[/itex].

I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this would be greatly appreciated.
 
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  • #2
Well clearly u is algebraic over K (it is an elemnt of K) so I'm guessing you mean to say that u is algebraic over F. Well if u^2 is algebraic over F then let f be a polynomial in F[x] such that f(u^2)=0. I don't want to give it away but if you think about the polynomial f(x^2)...
 
  • #3
Oh wow yea, ok it's pretty clear. I think I was complicating things. Thanks.
 

1. What is a simple field extension?

A simple field extension is a type of algebraic field extension in which the new field is generated by adjoining a single element to the original field. This means that the new field contains all elements of the original field as well as the new element.

2. How is a simple field extension different from a general field extension?

A simple field extension is a special case of a general field extension, where the new field is generated by a single element. In a general field extension, the new field can be generated by multiple elements.

3. What are some examples of simple field extensions?

One example of a simple field extension is the complex numbers, which are generated by adjoining the imaginary unit i to the real numbers. Another example is the field of algebraic numbers, which is generated by adjoining a root of a polynomial to the rational numbers.

4. Can a simple field extension be finite?

Yes, a simple field extension can be finite. For example, the finite field of 5 elements can be generated by adjoining a root of the polynomial x^2+x+1 to the integers modulo 5.

5. How are simple field extensions used in mathematics and science?

Simple field extensions are used in various areas of mathematics and science, including abstract algebra, number theory, and Galois theory. They also have applications in physics and engineering, particularly in the study of fields such as quantum mechanics and electromagnetism.

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