Galois extension of a field with Characteristic 0

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SUMMARY

The discussion focuses on the properties of Galois extensions of fields with characteristic 0, specifically addressing the existence of a unique smallest subfield K such that F ⊆ K ⊆ E, where K is normal over F and E is subradical over K. The key result utilized is that if A and B are solvable subgroups of a group G with A normal in G, then AB is also solvable. Additionally, the discussion explores the Galois group Gal(f) of an irreducible polynomial f over ℚ of degree 5 with at least two complex roots, concluding that its order must be one of 10, 20, 60, or 120.

PREREQUISITES
  • Understanding of Galois theory and extensions
  • Familiarity with solvable groups and their properties
  • Knowledge of irreducible polynomials and their Galois groups
  • Basic concepts of field theory, particularly in characteristic 0
NEXT STEPS
  • Study the structure of Galois groups, particularly for polynomials of degree 5
  • Explore the relationship between solvable groups and their normal subgroups
  • Investigate the concept of subradical extensions in field theory
  • Learn about the implications of the Fundamental Theorem of Galois Theory
USEFUL FOR

Mathematicians, particularly those specializing in algebra, field theory, and Galois theory, as well as graduate students seeking to deepen their understanding of Galois extensions and their properties.

bham10246
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This is something I've been trying to work on on my own for the past few days but I'm not sure how to approach it.

My Question:
a. Let E be a Galois extension of a field F with characteristic 0. Prove that there is a unique smallest subfield K such that [itex]F \subseteq K \subseteq E[/itex], K is normal over F and E is subradical over K.
[One needs the following result: Let [itex]A[/itex] and [itex]B[/itex] be solvable subgroups of a group [itex]G[/itex] and suppose that [itex]A[/itex] is normal in [itex]G[/itex]. Then [itex]AB[/itex] is solvable.]

b. Let f be an irreducible polynomial over [itex]\mathbb{Q}[/itex] which has degree 5 and at least two complex roots. Prove that [itex]Gal(f)[/itex] has order 10, 20, 60, or 120.

Thanks in advance for any kind of direction that you can provide me with...
 
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what does smallest mean? the intersectiuoin of all such thigns? or smallest degree?

and are your extensions of finite degree?

so i guess you are claiming there is a unique largest normal solvable subgroup of a group. your description majkes it fairly obvious how to proceed.
 

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