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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
is split over the rationals
Example
The three irreducible complex representations of are already defined over : the trivial representation, the sign representation, and the two-dimensional standard representation on by permutation of coordinates. Hence is a splitting field for .
Facts & Assumptions
Given: The natural coordinate-permutation action of on .
The number of irreducible complex representations of is its number of conjugacy classes, namely three (If is algebraically closed and , the number of irreducible representations of equals the number of conjugacy classes).
The trivial and sign representations are defined over every field of characteristic zero (The trivial representation, the regular representation, and permutation representations from finite -sets, The sign representation of and the restriction of a representation to a subgroup).
Verification
The line is trivial and its invariant complement has dimension two. A transposition has trace on , while a -cycle has trace ; thus is neither trivial nor sign.
The three displayed rational models have distinct complex characters, and [L1] says there are no further irreducibles. Therefore every complex irreducible has a rational model, which is exactly that is splitting for .
Depends on
- The finite symmetric group $S_n$, one-line notation, and cycle notation
- The sign representation of $S_n$ and the restriction $\operatorname{Res}^G_H(V)$ of a representation to a subgroup
- The trivial representation, the regular representation, and permutation representations from finite $G$-sets
- If $k$ is algebraically closed and $\operatorname{char} k \nmid |G|$, the number of irreducible representations of $G$ equals the number of conjugacy classes
- A splitting field for a finite group: every irreducible representation has scalar endomorphism ring
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
20 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
- Gabor Wiese, Galois Representations, Section 2.3 (standard reference, not scraped)