Question

any field is a subspace over a subfield​

Answer 1

Answer:

There is no such field F. Suppose otherwise. Choose any x∈R−F, so {1,x} is a basis of R over F. Since the degree of the extension equals 2, there is a polynomial x2+ax+b=0 with coefficients a,b∈F. By the quadratic formula, x is an F-linear combination of 1 and y√ for some y∈F and y must be positive since x is real. It follows that {1,y√} is a basis of R over F.

Consider z=y√4∈R, z>0. We can write z=a+by√ where a,b∈F. Solving for a,b in the equation z2=y√ we obtain a4=−14y. However, a negative number has no fourth roots in R, contradiction.

Answer 2

Answer:

I have this doubt, I can consider a field also a subspace? For example I have the vector space of polynomials with coefficients in K in the indeterminate x: K<=n . If n=0 I have K which is a field, so it is a vector subspace and in general every field is a subspace?

i hope this will helpful for u.

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