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2: Let E and F be extensions of a field K such that [E : K] < ∞. Then E and F are linearly disjoint over K if and only if [E : K] = [EF : F ]. If in addition [F : K] < ∞, then this is equivalent to [EF : K] = [E : K][F : K]. Proof: E and F are linearly disjoint over K and w1 , . . , wn is a basis for E/K, then w1 , . . , wn is also a basis for EF over F . Hence [EF : F ] = n = [E : K]. Conversely, suppose that [E : K] = [EF : F ] and let x1 , . . , xm ∈ E be linearly independent over K. Extend {x1 , .

Eg . If e1 = e2 = · · · = eg = 1, then p is unramified in L. If, in addition L/K is separable, then only finitely many ideals of R are ramified in L. Indeed, R has an ideal d, called the discriminant of S/R such that p is ramified in L if and only if p divides d. Assume in addition, that L is a finite Galois extension of K. e, P ∩ R = p). The decomposition group of P over K is the following subgroup of Gal(L/K): DP = {σ ∈ Gal(L/K) | Pσ = P}. The fixed field L(P) of D(P) in L is the decomposition field of P over K.

Eg . If e1 = e2 = · · · = eg = 1, then p is unramified in L. If, in addition L/K is separable, then only finitely many ideals of R are ramified in L. Indeed, R has an ideal d, called the discriminant of S/R such that p is ramified in L if and only if p divides d. Assume in addition, that L is a finite Galois extension of K. e, P ∩ R = p). The decomposition group of P over K is the following subgroup of Gal(L/K): DP = {σ ∈ Gal(L/K) | Pσ = P}. The fixed field L(P) of D(P) in L is the decomposition field of P over K.

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