When the Product of Non-Squares is a Square

[Kreizhn]

Homework Statement

For some field F, if a and b are two elements which are not squares but whose product is a square, show that there is an element k such that a = k^2 b.

The Attempt at a Solution

This is a boiled down version of what I'm actually trying to show, so the statement above may not actually be correct. If it's not, please let me know!

I have a few ideas of how to hit this, but the following is the one I like the most:

If ab = c^2 for some element c \in F then it is sufficient to show that either a | c or b | c, since then (without loss of generality) if a | c then there is some constant k such that ak = c in which case ab = a^2 k^2 which implies that b = k^2 a. Now the hard part (for me at least!) is showing that ab = c^2 implies that either a|c or b|c. It seems like this result should be true, but I've been having trouble showing it.

I start like this. If both a|c and b|c then we are done, hence assume without loss of generality that b \not| c. Here is where I get stuck. Any ideas?

 

[Bacle2]

Sorry to change your strategy, but , in the factoring of a square, every factor should
appear an even number of times.

Then ab=∏x_i^2ki

 

[Dick]

Homework Statement

For some field F, if a and b are two elements which are not squares but whose product is a square, show that there is an element k such that a = k^2 b.

The Attempt at a Solution

This is a boiled down version of what I'm actually trying to show, so the statement above may not actually be correct. If it's not, please let me know!

I have a few ideas of how to hit this, but the following is the one I like the most:

If ab = c^2 for some element c \in F then it is sufficient to show that either a | c or b | c, since then (without loss of generality) if a | c then there is some constant k such that ak = c in which case ab = a^2 k^2 which implies that b = k^2 a. Now the hard part (for me at least!) is showing that ab = c^2 implies that either a|c or b|c. It seems like this result should be true, but I've been having trouble showing it.

I start like this. If both a|c and b|c then we are done, hence assume without loss of generality that b \not| c. Here is where I get stuck. Any ideas?

Are you sure you are working in a FIELD? If that's the case, then you can always solve ak=c for k if a is nonzero. k=c*a^(-1).

 

[Kreizhn]

Sorry to change your strategy, but , in the factoring of a square, every factor should
appear an even number of times.

Then ab=∏x_i^2ki

Bacle2, I agree entirely. In particular, if we apply this to the case when a and b are not squares it implies that a and b must share a prime factor, since otherwise they would be squares themselves. However, I don't quite see why that helps. Care to elaborate?

Are you sure you are working in a FIELD? If that's the case, then you can always solve ak=c for k if a is nonzero. k=c*a^(-1).

Yes, you are of course correct. I wonder if this would actually work for what I was originally trying to do. I'll have to think about this.

In any case, I believe that the statement

"If ab = c^2 then either a|c or b|c"

is still correct. I would be interested in seeing a proof of this if anyone can suggest one.

 

[Dick]

Bacle2, I agree entirely. In particular, if we apply this to the case when a and b are not squares it implies that a and b must share a prime factor, since otherwise they would be squares themselves. However, I don't quite see why that helps. Care to elaborate?



Yes, you are of course correct. I wonder if this would actually work for what I was originally trying to do. I'll have to think about this.

In any case, I believe that the statement

"If ab = c^2 then either a|c or b|c"

is still correct. I would be interested in seeing a proof of this if anyone can suggest one.

The proof is trivial in a field. Everything divides everything. If you are working over the integers then 4*9=36=6^2. Neither 4 nor 9 divides 6.

 

[Kreizhn]

Ah excellent. Thank you.

 

[Bacle2]

While Dick gave you a nice answer, my idea was to use the fact that fields are UFD's, so that the product ab has a unique expression in which c must appear twice on the LH side.

 

[Dick]

While Dick gave you a nice answer, my idea was to use the fact that fields are UFD's, so that the product ab has a unique expression in which c must appear twice on the LH side.

How many fields have you seen that are UFD's??

 

[Bacle2]

Every field is a UFD, I'm almost 100%. Do you have a counter?

 

[Dick]

Every field is a UFD, I'm almost 100%. Do you have a counter?

The rationals, the reals, the complexes, integers modulo a prime p where p isn't 2, fields of order p^n where n>1, Galois extension fields. Do you want me to go on? Basically all of them. Think about it. What are the primes in the rationals, Q?

 

[Bacle2]

I'm talking about irreducibles.

If F is a field, then F[X] is a UFD, e.g: http://planetmath.org/encyclopedia/PrimeFactor2.html

This means that every poly. in F[X] has a unique factorization. Now,F embeds trivially

in F[X], right? Or isn't every f in F an element in F[X]?

Just use the embedding f--> f+ 0.x+0.x²+...+0xⁿ

It then follows every f in F has a unique factorization.

 

[Dick]

I'm talking about irreducibles.

If F is a field, then F[X] is a UFD, e.g: http://planetmath.org/encyclopedia/PrimeFactor2.html

This means that every poly. in F[X] has a unique factorization. Now,F embeds trivially

in F[X], right? Or isn't every f in F an element in F[X]?

Just use the embedding f--> f+ 0.x+0.x²+...+0xⁿ

It then follows every f in F has a unique factorization.

You seem to be googling instead of thinking. Unique factorization of polynomials is modulo the underlying field F. x^2-1=(x-1)(x+1)=(2x-2)(x/2+1/2). 2=3*(2/3)=5*(2/5). Does that look like unique factorization?

 

[Bacle2]

I was referring to your statement:

How many fields have you seen that are UFD's??

So every field is a UFD, by the definition of UFD, since, trivially, every element of a field is a unit. I just used the link to avoid doing a proof of this fact. Maybe you were
using the term in relation to this problem/situation which I guess I missed, but, unless otherwise stated, every field is a UFD.

I just want to add that sometimes I go thru old posts, and I prefer to be maybe overly-finicky in using terminology to avoid picking up things that , however answer
a question, are not fully correct in the strict sense.

 

[Dick]

While Dick gave you a nice answer, my idea was to use the fact that fields are UFD's, so that the product ab has a unique expression in which c must appear twice on the LH side.

Yes, everything is a unit. So it's a trivial UFD. This post is what I've actually been going on about. What is the 'unique expression'? 1?

 

[Bacle2]

My point is that once the expression is fixed, there is a unique, trivial solution, because every element divides every element, by the definition of field.

Then a|c is tautological for every non-zero element in the field, because every element is a unit, then a|c , inthe sense that a/c :=ac^-1 , which is in F by closure properties of F.

Sorry, my connection is not good tonight.

 

[Bacle2]

Anyway, this thread has become an exercise in futility and misunderstanding (misunderestimating?).

Yes, I guess I could have avoided bringing up the UFD issue, which does not really help, to say the least. My bad.


Let me delete my posts; they only seem to add confusion. Just wanted to ask.

 

[Dick]

Anyway, this thread has become an exercise in futility and misunderstanding (misunderestimating?).

Yes, I guess I could have avoided bringing up the UFD issue, which does not really help, to say the least. My bad.Let me delete my posts; they only seem to add confusion. Just wanted to ask.

You are right, of course. Not having any irreducible elements does NOT mean a field is not a UFD. Sorry, I seem to have gotten stuck in that impression. May have been too late.