Theorem 3.2
Let L/K be a finite separable extension of local fields. Then there exists a unique field L0 such that K subset L0 subset L and L0/K is unramified, L/L0 is totally ramified.
Proof:
We have Hom_K(L0, L) = Hom_kK(kL, kL) 
(question: why?)
and then L0 inj L (why? Hint: consider the id on the right hand side set).

Theorem 3.2 Let L/K be a finite separable extension of local fields. Then there exists a unique field L0 such that K subset L0 subset L and L0/K is unramified, L/L0 is totally ramified. Proof: We have Hom_K(L0, L) = Hom_kK(kL, kL) (question: why?) and then L0 inj L (why? Hint: consider the id on the right hand side set).

Theorem 3.2
Let L/K be a finite separable extension of local fields. Then there exists a unique field L0 such that K subset L0 subset L and L0/K is unramified, L/L0 is totally ramified.
Proof:
We have Hom_K(L0, L) = Hom_kK(kL, kL)&nbsp;
(question: why?)
and then L0 inj L (why? Hint: consider the id on the right hand side set).

Totally ramified extension (sketching proof) #ALNT-LB 2.3.SP Chapter 2 Section 3 #Algebraic Number Theory # Lecture note Benois

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Totally ramified extension (sketching proof) #ALNT-LB 2.3.SP Chapter 2 Section 3 #Algebraic Number Theory # Lecture note Benois

Totally ramified extension (sketching proof) #ALNT-LB 2.3.SP Chapter 2 Section 3 #Algebraic Number Theory # Lecture note Benois