Title: Daniel's Flashcards / Year 4 / Term 1 / Local Fields

html: Definition 1.8. The p-adic numbers ℚ p is the completion of ℚ with respect to | ∙ | p.

html: Definition 2.2. Let ( K , | ∙ | ) be a non-archimedean valued field. The valuation ring of K is defined to be
O K = { x ∈ K | | x | ≤ 1 } ( = B ¯ ( 0 , 1 ) ) ( = { x ∈ K × | v ( x ) ≥ 0 } ∪ { 0 } )

html: Definition 2.5. Let v : K × → ℝ be a valuation. If v ( K × ) ≅ ℤ, we say v is a discrete valuation. K is said to be a discretely valued field. An element π ∈ O K is uniformiser if v ( π ) > 0 and v ( π ) generates v ( K × ).

html: Definition 3.1. The ring of p-adic integers ℤ p is the valuation ring
ℤ p = { x ∈ ℚ p | | x | p ≤ 1 } .

html: Definition 5.3. The element [ a ] ∈ O K constructed in Theorem 5.2 is the Teichmüller lift of a.

html: Definition 14.4. G 1 is called the wild inertial group, and G 0 ∕ G 1 is called the tame quotient.

html: Definition 15.2. Let ( I , ≤ ) be a directed set and ( G i ) i ∈ I a collection of groups together with maps φ i j : G j → G i, i ≤ j such that:
- φ i k = φ i j ∘ φ j k for any i ≤ j ≤ k
- φ i i = id ⁡
Say ( ( G i ) i = 1 , φ i j ) is an inverse system. The inverse limit of ( G i , φ i ) is
lim ⁡ i [ ] ← G i = { ( g i ) i ∈ I ∈ ∏ i ∈ I G i | φ i j ( g j ) = g i } .