Some true or false Field extension theory questions

In summary, the extension ##F=\mathbb{Q}(S)## is algebraic, infinite, not simple, and separable. It can be seen as a simple extension by multiplying all minimal polynomials, but this only works for finite extensions. The extension is also separable due to the characteristic of ##\mathbb{Q}## being 0.
  • #1
PsychonautQQ
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Homework Statement


Let S= {e^2*i*pi/n for all n in the natural numbers} and let F=Q

Is F:Q
1) algebraic?
2) finite?
3) simple?
4)separable?

Homework Equations

The Attempt at a Solution


1) Every element in S is a root of x^n-1 and every element of a in Q is a root of x-a, and thus I think somehow that means that the whole extension is algebraic is (i.e all the basis elements are algebraic and obviously all the elements of the base field are algebraic). Does this argument work? If not what would?

2) No. Every p in the natural numbers will be a basis that is linearly independent of all the elements before it, and there are infinite primes, so this will not be a finite extension.

3) I'm not sure. I want to say there was some theorem that algebraic extensions can ultimately be seen as simple but I'm not sure.

4) Yes, this extension is over Q where char(Q) = 0 so the extension is separable.
 
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  • #2
PsychonautQQ said:

Homework Statement


Let S= {e^2*i*pi/n for all n in the natural numbers} and let F=Q

Is F:Q
I guess you mean ##F=\mathbb{Q}(S)##.
1) algebraic?
2) finite?
3) simple?
4)separable?

Homework Equations

The Attempt at a Solution


1) Every element in S is a root of x^n-1 and every element of a in Q is a root of x-a, and thus I think somehow that means that the whole extension is algebraic is (i.e all the basis elements are algebraic and obviously all the elements of the base field are algebraic). Does this argument work? If not what would?
Yes. Although you don't need to mention the elements of ##\mathbb{Q}## itself. E.g. all real numbers are trivially algebraic over ##\mathbb{R}##, even if not over ##\mathbb{Q}##. Algebraic over itself is trivial, but your argument is correct.
2) No. Every p in the natural numbers will be a basis that is linearly independent of all the elements before it, and there are infinite primes, so this will not be a finite extension.
Yes. I don't think they are already a basis, because their powers until ##p-1## need to be included as well. However, they are linear independent and this is sufficient here.
3) I'm not sure. I want to say there was some theorem that algebraic extensions can ultimately be seen as simple but I'm not sure.
How could this be done with infinitely many elements in ##S##? Read the theorem again, I'm sure it states something like: Every finite algebraic extension ... As far as I remember, the argument here is to simply multiply all minimal polynomials.
4) Yes, this extension is over Q where char(Q) = 0 so the extension is separable.
Not 100% sure whether separability applies to infinite extensions, too, but you're right. No multiple roots.
 
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FAQ: Some true or false Field extension theory questions

1. What is a field extension?

A field extension is a mathematical concept that involves extending a field (such as the rational numbers or real numbers) to include new elements that satisfy certain algebraic properties.

2. What is the difference between a finite and infinite field extension?

A finite field extension is one in which the number of elements in the extended field is finite, while an infinite field extension has an infinite number of elements. In other words, a finite field extension has a finite degree, while an infinite field extension has an infinite degree.

3. Can a field have more than one extension?

Yes, a field can have multiple extensions, each with different properties and characteristics. For example, the field of rational numbers has extensions such as the real numbers and the complex numbers.

4. What is the degree of a field extension?

The degree of a field extension is the number of elements that are added to the original field in the extension process. It is also equal to the dimension of the extension as a vector space over the original field.

5. How is the Galois group related to field extensions?

The Galois group is a fundamental concept in field extension theory, which describes the symmetries and automorphisms of an extension field. It helps to understand the properties and structure of the extension, and is useful in solving problems related to field extensions.

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