Linking Invariants for Valuations and Orderings on Fields
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914952282898432 |
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| author | Efrat, Ido |
| author_facet | Efrat, Ido |
| contents | The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of $\mathbb{Q}$ with unitriangular Galois groups, and contain the Legendre and Redei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the q-th roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient, and relate it to Massey product elements in Galois cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linking Invariants for Valuations and Orderings on Fields Efrat, Ido Number Theory Primary 11R32, Secondary 11R34, 12G05, 12J10 The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of $\mathbb{Q}$ with unitriangular Galois groups, and contain the Legendre and Redei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the q-th roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient, and relate it to Massey product elements in Galois cohomology. |
| title | Linking Invariants for Valuations and Orderings on Fields |
| topic | Number Theory Primary 11R32, Secondary 11R34, 12G05, 12J10 |
| url | https://arxiv.org/abs/2403.07482 |