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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Number-field discriminant is well-defined and nonzero
Statement
Integral bases give the same nonzero signed integer .
Facts & Assumptions
Given: Two integral bases of .
Change of basis squares its determinant (Change of basis for discriminants).
The embedding determinant formula holds (Embedding determinant formula).
Proof
The integral change matrix is unimodular, so [F1] gives basis independence; integral traces make the value integral.
Distinct embeddings make the matrix in [F2] invertible, so its square is nonzero.
Depends on
Used by
- Order-index discriminant formula Corollary
- Integers of Q Example
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
- Milne, Proposition 2.27 (standard reference, not scraped)