Question:-
Show that a field K can be regarded as a
vector space over any subfield F of K.
Solution:- Let K be a field and F be its subfield.
Since, K
is a field,
Therefore,
{2}- Let a∊F ,α∊k ⇨ aα∊K (Since ,a∊F ⇨ α∊K)
Therefore
, a∊K, α∊K ⇨ aα∊K
Hence, K
is closed under the operation of multiplication of K.
{3}- Let a,b∊F, α,β∊K
Since, a,
b∊K and α,β∊K as F⊆K
(1)- a(α+β)=aα+bβ (by left distribution law of K)
(2)- (a+b)α=aα+bα (by right distribution law of K)
(3)- (ab)α=a(bα) (by associativity of multiplication)
(4)- 1.α=α, where 1∊F & α∊K
Hence K
satisfies all properties of a vector space over its subfield F. Thus K(F) is a
vector space.
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