Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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FALSE: the Galois correspondence preserves inclusion
Statement
False claim. If are subgroups in a finite Galois correspondence, then .
Facts & Assumptions
Given: The general inclusion-reversing correspondence of The fundamental theorem of finite Galois theory.
The trivial subgroup fixes the whole biquadratic extension, while each order-two subgroup fixes a quadratic field (The complete Galois correspondence for ).
Refutation
In [L1], the trivial subgroup is strictly contained in an order-two subgroup, but its fixed field is the entire biquadratic field and strictly contains the quadratic fixed field of the larger subgroup.
The subgroup containment in step 1.1 produces the reverse strict field containment, so it refutes inclusion preservation.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- J. S. Milne, Fields and Galois Theory, v5.10, Theorem 3.17 (standard reference, not scraped)
- K. Conrad, The Galois Correspondence, Theorem 5.6 (standard reference, not scraped)