Question about fields and vector spaces

Question: Suppose K and L are fields and $\displaystyle K\subset L$. Show that L is a vector space over K.

Firstly, I'm not quite sure what it means to say "L is a vector space over K" and secondly, I'm not quite sure how to show that L is a vector space. Help please?

Re: Question about fields and vector spaces

Hint. The set of real numbers is a vectorspace over its self, can you find a subfield Kof R that lets (R, K) into a vector space? I'm sure you can. Use this idea in your general case.

Re: Question about fields and vector spaces

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**worc3247** Question: Suppose K and L are fields and $\displaystyle K\subset L$. Show that L is a vector space over K.

Firstly, I'm not quite sure what it means to say "L is a vector space over K" and secondly, I'm not quite sure how to show that L is a vector space. Help please?

What it is asking you to prove that if you define the multiplication of an element of $\displaystyle K$ and an element of $\displaystyle L$ in the normal way (the way the field says so) then this makes $\displaystyle L$ into a $\displaystyle K$-space.

Re: Question about fields and vector spaces

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Originally Posted by

**worc3247** Question: Suppose K and L are fields and $\displaystyle K\subset L$. Show that L is a vector space over K.

Firstly, I'm not quite sure what it means to say "L is a vector space over K" and secondly, I'm not quite sure how to show that L is a vector space. Help please?

You can't do any math problems if you don't know the **definitions**! What is the definition of "V is a vector space over field K"?

Re: Question about fields and vector spaces

just in case you have forgotten, here is the definition:

a vector space over a field F is a set V, together with two operations: +:VxV--->V and *:FxV-->V such that:

1.(V,+) is an abelian group

2. for c in F, and u,v in V, c*(u+v) = c*u + c*v

3. for c,d in F, and u in V, (c+d)*u = c*u + d*u

4. for c,d in F, and u in V, c*(d*u) = (cd)*u

5. for u in V, and the (multiplicative) identity 1 of F, 1*u = u.

for a subfield K of a field L, we may take F = K and V = L (yes, L has additional structure, but that is beside the point).

the vector + operation is defined to be the addition of L.

the scalar multiplication is defined to be k*l = kl, the product of k and l in the field L.

now, verify the axioms 1 through 5.

(the normal example for this kind of thing is the complex field, and the real field, where the complex field is a 2 dimensional vector space over the reals, with basis {1,i}).

Re: Question about fields and vector spaces

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**HallsofIvy** You can't do any math problems if you don't know the **definitions**! What is the definition of "V is a vector space over field K"?

With or without definitions, one cannot solve an ambiguously stated problem. A vector space is a tuple of which V is only one element. Another element of this tuple is the multiplication of elements of the field K and elements of V. What this multiplication is supposed to be isn't said in the problem, although it is rather obvious. Actually, other operations of the space aren't defined either.

I don't think I've seen this problem formulated unambiguously though.

Re: Question about fields and vector spaces

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**ymar** With or without definitions, one cannot solve an ambiguously stated problem. A vector space is a tuple of which V is only one element. Another element of this tuple is the multiplication of elements of the field K and elements of V. What this multiplication is supposed to be isn't said in the problem, although it is rather obvious. Actually, other operations of the space aren't defined either.

I don't think I've seen this problem formulated unambiguously though.

that is a good point. it is possible some other operation might be intended than just the (given) field multiplication in L. and the vector sum isn't explicitly stated, either.

a better way to phrase the question would be:

"Prove that if K is a subfield of a field L, that L is a vector space over K, with vector addtion being the field addition of L, and scalar multiplication being the field multiplication of L". (being well-defined since K is entirely contained within L).

because given even just an abelian group, it is possible to define other operations on that set and wind up with a different abelian group with the same underlying set.

that said, i don't think the ambiguity of the wording of the problem, is what is giving the original poster his difficulty.

Re: Question about fields and vector spaces

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**Deveno** that said, i don't think the ambiguity of the wording of the problem, is what is giving the original poster his difficulty.

I agree, although I can imagine it being troublesome. I've been there. It's not easy when you have to cope with completely new concepts defined in abstract ways. I still find myself spewing out profanity at the authors of ambiguous sentences in math books and later wondering how I could have any trouble with them. Sometimes an hour later. :)

Re: Question about fields and vector spaces

Ok thanks, I think I've got it.

Re: Question about fields and vector spaces

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**worc3247** Ok but unless K is the reals, how can you you have scalar multiplication?

To quote HallsOfIvy, "You can't do any math problems if you don't know the **definitions**!"

so, what is the definition of a Field?

(My point is that all fields have scalar multiplication!)

Re: Question about fields and vector spaces

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Originally Posted by

**Swlabr** To quote HallsOfIvy, "You can't do any math problems if you don't know the **definitions**!"

so, what is the definition of a Field?

(My point is that all fields have scalar multiplication!)

Yes I got it now :) thanks!

Re: Question about fields and vector spaces

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Originally Posted by

**Swlabr** To quote HallsOfIvy, "You can't do any math problems if you don't know the **definitions**!"

so, what is the definition of a Field?

(My point is that all fields have scalar multiplication!)

well, the "field" with 1 element (the so called "F-un") doesn't

(note to original poster, and others: this is a mathematical in-joke, there is no field with 1 element. but if there were....)

Re: Question about fields and vector spaces

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Originally Posted by

**Deveno** well, the "field" with 1 element (the so called "F-un") doesn't

(note to original poster, and others: this is a mathematical in-joke, there is no field with 1 element. but if there were....)

Interestingly, it was Tits who contributed to the discussion of F-un.