Galois Extensions: Homework Analysis

  • Thread starter shoplifter
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In summary, the conversation discusses the concept of Galois extensions and whether certain fields are Galois over other fields. The participants also consider conditions for a field to be Galois and attempt to find explicit polynomials that would prove or disprove Galois extensions. However, it is ultimately concluded that the last field mentioned (\mathbb{Q}(\beta + i\beta)/\mathbb{Q}(\sqrt{-5})) is Galois over \mathbb{Q}(\sqrt{-5}) due to the polynomial x^2 - 2i*b^2. The status of the other two fields (\mathbb{Q}(\beta + i\beta)/\mathbb{Q} and \mathbb{Q
  • #1
shoplifter
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Homework Statement



Let [tex]\beta[/tex] be a real, positive fourth root of 5. Is [tex]\mathbb{Q}(\sqrt{-5})[/tex] Galois over [tex]\mathbb{Q}[/tex]? How about [tex]\mathbb{Q}(\beta + i\beta)/\mathbb{Q}[/tex] or [tex]\mathbb{Q}(\beta + i\beta)/\mathbb{Q}(\sqrt{-5})[/tex]?


Homework Equations



An extension is Galois when it is normal and separable.

The Attempt at a Solution



Any push whatsoever in the right direction would be much appreciated.
 
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  • #2
Can you tell me anything at all about those extensions?
 
  • #3
Do you know a condition equivalent to being Galois that involves splitting?
 
  • #4
well, a field [tex]A[/tex] is galois over [tex]B[/tex] if it is the splitting field of some polynomial in [tex]B[x][/tex] which has distinct roots in [tex]A[/tex].

so then, [tex]\mathbb{Q}(\sqrt{-5})[/tex] is the splitting field of the polynomial [tex]x^2 + 5[/tex], which has coefficients in [tex]\mathbb{Q}[/tex]. this is because [tex]x^2 + 5 = (x + \sqrt{-5})(x - \sqrt{-5})[/tex], and hence the roots are also distinct, and so in the first case, we do have a galois extension.

but i can't apply this to the second two ideas, because i can't tell anything about the field [tex]\mathbb{Q}(\beta + i\beta)[/tex]. it doesn't seem to be the same as [tex]\mathbb{Q}(i, \beta)[/tex] or anything of the sort. am i missing something obvious?

thanks so much.
 
  • #5
also, thinking about the polynomial [tex](x + \beta + i\beta)(x + \beta - i\beta)(x - \beta + i\beta)(x - \beta - i\beta)[/tex] doesn't help, because it gives me the information that [tex]\mathbb{Q}(\beta + i\beta)[/tex] is galois over [tex]\mathbb{Q}(\sqrt{5})[/tex], but nothing else, and i don't see how this helps in the problem.
 
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  • #6
shoplifter said:
also, thinking about the polynomial [tex](x + \beta + i\beta)(x + \beta - i\beta)(x - \beta + i\beta)(x - \beta - i\beta)[/tex] doesn't help, because it gives me the information that [tex]\mathbb{Q}(\beta + i\beta)[/tex] is galois over [tex]\mathbb{Q}(\sqrt{5})[/tex], but nothing else, and i don't see how this helps in the problem.

Is b-ib really in Q(b+ib)?
 
  • #7
no, it isn't, but it doesn't follow that it's not galois, right? I've worked for a while, and can't find explicit separable polynomials whose splitting fields are precisely those, but then what does that prove?

i really don't see a way to disprove they're galois extensions. even if they are, i can't find the explicit polynomials, but my intuition points to a disproof. =( but how do i do that?
 
  • #8
oh i got the fact that it's galois over the third one -- from the polynomial x^2 - 2i*b^2.

what about the last one?
 

FAQ: Galois Extensions: Homework Analysis

1. What is a Galois extension?

A Galois extension is a type of field extension in abstract algebra, named after the mathematician Évariste Galois. It is a finite field extension where the extension field has a Galois group that is isomorphic to the automorphism group of the base field.

2. What is the significance of Galois extensions?

Galois extensions are important in the study of field theory and algebraic number theory. They provide a way to understand the structure and properties of fields, and have applications in many areas of mathematics, including cryptography and coding theory.

3. How do I determine if a field extension is Galois?

A field extension is Galois if and only if it is normal and separable. Normality means that the extension field contains all the roots of irreducible polynomials in the base field, and separability means that these roots are distinct. These conditions can often be checked using the Galois correspondence or by calculating the discriminant of the extension.

4. What is the Galois correspondence?

The Galois correspondence is a fundamental result in Galois theory that establishes a one-to-one correspondence between subfields of a Galois extension and subgroups of its Galois group. This correspondence allows us to use group theory to understand the structure and behavior of field extensions.

5. Are there any practical applications of Galois extensions?

Yes, Galois extensions have many practical applications in mathematics, engineering, and computer science. For example, they are used in the design of error-correcting codes, encryption algorithms, and in solving polynomial equations. They also have connections to other areas of mathematics such as topology and algebraic geometry.

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