Galois Solvable Group and Realizability over \mathbb{Q} for |G|=p^n

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Discussion Overview

The discussion revolves around the realizability of Galois groups over the rational numbers \(\mathbb{Q}\), specifically focusing on groups of order \(p^n\) and their relationship to solvable groups and the Inverse Galois Problem.

Discussion Character

  • Debate/contested, Technical explanation

Main Points Raised

  • One participant suggests that all solvable groups are known to be Galois groups, referencing Shafarevich's work.
  • Another participant questions the value of Shafarevich's theorem in the context of the Inverse Galois Problem, implying that its significance may diminish if the problem is true.
  • A later reply defends Shafarevich's contributions, arguing that they represent significant progress toward the Inverse Galois Problem.
  • There is a request for a PDF version of the referenced materials, indicating interest in further reading.
  • One participant inquires whether the realizability issue is resolved for abelian extensions.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Shafarevich's work and the Inverse Galois Problem, indicating a lack of consensus on the significance of these contributions to the discussion of Galois groups.

Contextual Notes

There are unresolved assumptions regarding the relationship between solvable groups and Galois groups, as well as the implications of the Inverse Galois Problem on existing theorems.

Kummer
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Galois Group

Is [tex]G[/tex] realizable over [tex]\mathbb{Q}[/tex] given that [tex]|G|=p^n[/tex] ?
 
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i think so. arent all solvable groups known to be galois groups? what did shafarevich prove?

http://www.math.uiuc.edu/Algebraic-Number-Theory/0136/
 
Is it possible to have this in pdf? Thank you.

(Note: Shafarevich's theorem and work will be worthless if the Inverse Galois Problem is true. (Unless it depends breaking the group into solvable groups first)).
 
your note strikes me as odd. i would say shafarevich's work is more accurately described as the best work so far toward the inverse galois problem.

see the book of serre, topics in galois theory.
 
Last edited:
Kummer said:
Is it possible to have this in pdf? Thank you.

(Note: Shafarevich's theorem and work will be worthless if the Inverse Galois Problem is true. (Unless it depends breaking the group into solvable groups first)).

Get a (free) ps viewer, or run ps2pdf (standard *nix program, and installed if you have LaTeX on Win).

Your last comment seems to be yet another of your over-arching and dismissive comments about mathematics. These are strange since you seem to know a lot of number theory. How can you dismiss this work as being worthlesss if IGP is true? Surely you must then think all mathematics is worthless if it doesn't prove absolutely everything simultaneously?
 
Is it solved for abelian extensions?
 

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