On ramification index of composition of complete discrete valuation fields
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On ramification index of composition of complete discrete valuation fields PASUPULATI SUNIL KUMAR Indian Institute of Science Education and Research, Thiruvananthapuram 695 551, India E-mail: [email protected] MS received 8 November 2019; revised 29 January 2020; accepted 10 February 2020 Abstract. For an extension L/K of discrete valuation fields, let e L/K , O L denote the ramification index and valuation ring of L/K respectively. Let K be a complete discrete valuation field and L 1 /K , L 2 /K be finite linearly disjoint extensions over K . We show that if O L 1 L 2 = O L 1 O L 2 or gcd(e L 1 /K , e L 2 /K ) = 1, and one of the residue fields l1 /k, l2 /k is separable, then e L 1 L 2 /L 1 = e L 2 /K . The analogous results for the residue degrees are also true. Keywords. extensions.
Complete discrete valuation fields; ramification indices; linearly disjoint
Mathematics Subject Classification.
Primary: 11S; Secondary: 11S15, 11S20.
1. Introduction An extension E ⊃ K is an extension of discrete valuation fields if the discrete valuation ω on E is equivalent to an extension of the discrete valuation υ on K . The ramification e index e E/K of E/K is defined by π K = π EE/K , where π K , π E are uniformizers of K and E respectively. Let F ⊂ K ⊂ L be extensions of complete discrete valuation fields. It is a well-known fact that e L/F = e L/K e K /F . The residue degree f E/K of the extension E/K is defined by [ (πOEE ) : (πOKK ) ], where O E , O K are valuation ring E and K respectively . Let L 1 , L 2 be finite extensions over a complete discrete valuation field K . Extend the discrete valuation υ on K to the discrete valuations ω1 and ω2 over L 1 and L 2 respectively. The e1 , e2 , e1∗ , e2∗ , e in Figure 1 indicates ramification indices of the corresponding extensions. We know that e = e1 e1∗ = e2 e2∗ , hence e = e1 e2 ⇔ e1 = e2∗ ⇔ e2 = e1∗ . In this article, we show that under certain conditions on L 1 /K and L 2 /K , e = e1 e2 . We know that L 1 , L 2 are linearly disjoint and finite extensions over K if and only if the dimension of the composite field L 1 L 2 of L 1 and L 2 is the product of the dimensions of L 1 and L 2 over K . That is, [L 1 L 2 : K ] = [L 1 : K ][L 2 : K ]. We also have that if L 1 /K , L 2 /K are complete discrete valuation fields, then [L 1 L 2 : K ] = e L 1 L 2 /K f L 1 L 2 /K = e L 1 /K f L 1 /K e L 2 /K f L 2 /K ([3], Chapter 2, Section 4). The natural question is whether the ramification indices can be equal on both sides and also whether the residue degrees can be equal on both sides. Our main purpose in this paper is to show that under certain © Indian Academy of Sciences 0123456789().: V,-vol
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Figure 1. Ramification indices of L 1 , L 2 , L 1 L 2 over K .
assumptions, e L 1 L 2 /K = e L 1 /K e L 2 /K and f L 1 L 2 /K = f L 1 /K f L 2 /K when L 1 /K , L 2 /K are linearly disjoint over K . Remark 1. Linearly disjoint assumption is must in our theorems. Because, if L 1 /K and L 2 /K are not linearly di
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