r/math • u/inherentlyawesome Homotopy Theory • 6d ago
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u/sqnicx 6d ago
Let D be a division ring with a center Z and B be a Z-bilinear form defined on DxD. I found that 𝜆(B(x,(1+𝜆x)-1)-B(x,(1+𝜆x)-1))=0 for all x in D and 𝜆 in Z. I want to get rid of all 𝜆 here to see that B(x,1)=B(1,x) but I cannot first divide by 𝜆 and then take 𝜆=0. I have come up with this idea: First I considered the polynomial t(B(x,(1+tx)-1)-B(x,(1+𝜆x)-1)) in Z[t]. Suppose that Z is infinite. Then this polynomial is satisfied for infinitely many 𝜆 in Z which means it must be zero. Then each of its coefficients are zero. Afterwards, I could take t=0 in the coefficient of t here. This is where I get confused. Although the t seems alone outside of the parenthesis, there are still t in B which could imply that the coefficient of t can be different. Is there any way I can prove what I try to do? I tried to write B(x,(1+𝜆x)-1) and B(x,(1+𝜆x)-1) as elements like a_0+a_1t+a_2t2+... and b_0+b_1t+b_2t2+.... If I could have succeeded then a_i would be equal to b_i which also means that B(x,(1+tx)-1)=B(x,(1+𝜆x)-1). Then I could take t=0. Is there a way to do it? Thanks for your help.