1. Question about fields and vector spaces

Question: Suppose K and L are fields and $K\subset L$. 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?

2. 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.

3. Re: Question about fields and vector spaces

Originally Posted by worc3247
Question: Suppose K and L are fields and $K\subset L$. 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 $K$ and an element of $L$ in the normal way (the way the field says so) then this makes $L$ into a $K$-space.

4. Re: Question about fields and vector spaces

Originally Posted by worc3247
Question: Suppose K and L are fields and $K\subset L$. 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"?

5. 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}).

6. Re: Question about fields and vector spaces

Originally Posted by 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.

7. Re: Question about fields and vector spaces

Originally Posted by 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.

8. Re: Question about fields and vector spaces

Originally Posted by 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.

9. Re: Question about fields and vector spaces

Ok thanks, I think I've got it.

10. Re: Question about fields and vector spaces

Originally Posted by 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!)

11. Re: Question about fields and vector spaces

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!

12. Re: Question about fields and vector spaces

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....)

13. Re: Question about fields and vector spaces

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.