I agree with Mr.Farid .

A quadratic form can notbe decomposed in a product of two linear forms

Thank you Dear all. Yes I guessed, but I need a concrete example and Farid gave it. Thank you again dear Farid, Jack Son and dear Peter.

I agree with Mr.Farid

To tackle the problem in general we must first define precisely what we mean by a quadratic form. On a vector space V of dimension n over a field K of characteristic not 2, a quadratic form Q(.) is characterized in a unique way by its associated symmetric bilinear form B(.,.), which is such that 2B(x,y)=Q(x+y)-Q(x)-Q(y). Recall that the "radical" Rad (V)=Rad(V,B) is then defined as the subspace consisting of the vectors x in V s.t. B(x,y)=0 for all y in V. There are 2 extreme cases: Rad(V)=V, in which case B is identically 0 ; and Rad(V)=0, in which case B is called "non degenerate".

Back to your problem, suppose that B is the product of 2 linear forms, i.e. B(x,y)=L1 (x)L2 (y). The point will be the determination of the dimension of Rad(V), which is here the subspace generated by the kernels Hi of Li . Each Hi is a hyperplane (of dimension n-1) of V, hence Rad(V) is not 0, i.e. V is degenerate. Besides, it is classically known from linear algebra that H1=H2 iff L1 and L2 are proportional. Thus Rad(V)=V (resp. is a hyperplane H) iff L1 and L2 are not proportional (resp. are proportional). In the first case B is 0 and there is nothing to add. In the second case, incorporating the factor of proportionality into the linear form, we can write B(x,y)=L(x)L(y) for all x,y in V, where the kernel of L is a hyperplane H. Introduce the quotient vector space V=V/H, which is a line (i.e. has dimension 1), generated say by the class a of a vector a in V, not in H. Denoting by x in V the class of x in V, we can define Q(x)=Q(x) because H=Rad(V), and L(x)=L(x) because H=ker(L), so that Q(x)=L(x)2 . Choosing a to be a basis of V , we can write the quadratic form as Q(x)=k2 Q(a), k in K, or B(x,x')=kk' Q(a). Conversely, knowing H=Rad(V), such a Q or B can obviously be lifted from V to V.

In conclusion : on V , a non null symmetric bilinear form B is the product of 2 linear forms L1 and L2 iff these are proportional and Rad(V,B)=ker(Li ).

@ Peter Breuer

Dear Peter,

I hope that my mathematical answer will contribute to calm you down.

Dear Hanifa,

Previous answers are helpful and reply your request. For more details, I recommend the following paper.

https://www.math.tecnico.ulisboa.pt/~ggranja/manuel.pdf

Best regards

@Issa Kaddoura

I don't see what the reference you recommend brings to Hanifa's question. Besides, the article deals only with quadratic forms over very particular fields or fields.

@Thong,

Notice that you answer started with an introduction to Quadratic Forms, the recommended article explains the details.

Also, Peter answer connects factorization of the quadratic forms with the roots ( single or multivariable) over a given field.

I like this idea; the recommended article contains a detailed example (Example 2.17. page 10).

It is related to peter"s answer and not Hanifa's question.

Let me remind you that I said: "all your answers were helpful, accommodate the request of Hanifa."

Best regards

Thank you Dear all, from Farid to Issam, Really your answers are helpful. Farid gave an example to support my guess. Peter connects his answer with roots (really I want to know, but I couldn't ask him more details because of his nervousness). Thank you Issam for the answer. The answer of Thong is super, exactly what I want to know: the sufficient conditions for factorization..

1) @Issam , @Thong

I am writing a course of bilinear and quadratic forms for students of second year graduation, so I want an answer which is suitable for their level. The article used tools of number theory, it is very far from them. The answer of Thong will help a lot it contains only linear algebra and tools of the subject itself.

2)

@Peter

Thank you dear Peter for details. I am happy that you are not nervous, so I can always ask you and benefit of your answers :)

3) The book I am writing is in French, I didn't finish yet, May be in two months it will be ready. When I finish I wish if any of you can check it and write something ( words in the preface, comments, suggestions,...)

Dear Hanifa, The best introduction to the algebraic theory of quadratic (or rather bilinear) forms that I know is contained in Emil Artin's book "Geometric Algebra" (the title is a private joke). Thre exists a french edition (Gauthier-Villars) , translated by Michel Lazard.

The answer yes, but not in general.

@ Muhamammad,

Could you please explain, when yes and when no?

Or an example of each case is appreciated.

@ Muhamammad

Your answer cannot be considered as a math. one. In my post of 2 months ago, I gave a necessary and sufficient condition. By definition of a NSC, it cannot be improved, unless you give an equivalent condition which sheds a new light (= a new view point). Can you do this ?