Prove or disprove that for every field F of characteristic two, there exist vector spaces U and V over F and mapping from U to V, which is F-homogeneous but not additive.
Dear @Salah Hamad,
Thank you very much for the expression of your opinion. Could you please clarify what does your remark refers to? Is it the question in the heading or the statement suggested to proving or disproving?
The following content may be helpful
https://math.stackexchange.com/questions/2132215/a-real-function-which-is-additive-but-not-homogenous
Dear Muhammad Arshad, thank you very much for your answer. You are absolutely right that in case you mention finding a mapping that is additive but not homogeneous is simple; I could even say that this is very simple. But my question was about the scalar field of characteristic two. Can you say something in this case?
The question was asked here September 28, 2020, gathered 878 reads and remained unanswered. Now I am happy to give the full answer with proof. From the point of view of this answer something in my question may seem strange. So, a few words before I give the answer.
For the moment when I posed my question, the proof I had was suitable for all fields but those of characteristic two. That is why my question was about these fields. But recently, Greg Oman from U Colorado suggested me not forget the general case. What happened next was what Georg Polya called the inventor's paradox. The more general problem appeared to be easier. Have a look at the solution.
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