Finite extensions of local fields¶
The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Core Idea¶
Finite extensions of local fields is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K. In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.
Scope of Application¶
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Unramified extension. Let L/K be a finite Galois extension of nonarchimedean local fields with finite residue fields \ell/k and Galois group G .
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Unramified extension. (ii) \mathcal{O}L / \mathfrak{p}\mathcal{O}L is a field, where \mathfrak{p} is the maximal ideal of \mathcal{O}K .
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Unramified extension. (v) If \pi is a uniformizing element of K , then \pi is also a uniformizing element of L .
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Unramified extension. When L/K is unramified, by (iv) (or (iii)), G can be identified with \operatorname{Gal}(\ell/k) , which is finite cyclic.
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Unramified extension. The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Clarity¶
A clear use of Finite extensions of local fields names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Manages Complexity¶
Finite extensions of local fields compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—in algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.—and the practical consequence—the above implies.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
- Check operation and conditions. Let L/K be a finite Galois extension of nonarchimedean local fields with finite residue fields \ell/k and Galois group G .
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Finite extensions of local fields transfers literally when a new case preserves the same carrier type, relation, and recognition test. Let L/K be a finite Galois extension of nonarchimedean local fields with finite residue fields \ell/k and Galois group G . (ii) \mathcal{O}L / \mathfrak{p}\mathcal{O}L is a field, where \mathfrak{p} is the maximal ideal of \mathcal{O}K . Beyond the home domain. No canonical parent is asserted for Finite extensions of local fields.
Relationships to Other Abstractions¶
Current abstraction Finite extensions of local fields Domain-specific
Parents (1) — more general patterns this builds on
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Finite extensions of local fields is a kind of Field Extension Domain-specific
Finite extensions of local fields satisfies the defining boundary of Field Extension: A field extension is a pair of fields L/K together with an embedding identifying K as a subfield of L, so that L becomes a vector space and algebra over K and can be studied by degree, generators, algebraicity, separability, normality, and automorphisms.
Hierarchy path (1) — routes to 1 parentless root
- Finite extensions of local fields → Field Extension
Neighborhood in Abstraction Space¶
Finite extensions of local fields sits in a crowded region of the domain-specific corpus (40th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Algebraic Structures, Groups & Operators (42 abstractions)
Nearest neighbors
- Local Analysis — 0.89
- Group Ring — 0.88
- Local class field theory — 0.87
- Classifying space for SO(n) — 0.87
- Tensor product of fields — 0.87
Computed from structural-signature embeddings · 2026-10-08