Alphabeta MathHow statement and proof provenance work
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FALSE: every subgroup in the Galois correspondence gives a normal subextension
Statement
False claim. Every subgroup of the Galois group of a finite Galois extension corresponds to an intermediate field normal over the base.
Facts & Assumptions
Given: The exact normality criterion of Normal subgroups, conjugate fields, and quotient groups in the Galois correspondence.
The three order-two subgroups correspond to three cubic fields that are not normal over (The full correspondence for the splitting field of ).
Refutation
Choose any order-two subgroup from [L1]. It is a subgroup in the finite Galois correspondence, but it is not normal in , and its fixed cubic field is not normal over .
The strict cubic fixed field in step 1.1 is therefore a counterexample to the universal normality claim.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- K. Conrad, The Galois Correspondence, Theorem 5.6 and Example 5.8 (standard reference, not scraped)