A polynomial of degree n in a field F has at most n roots.
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LoginNumber of elements in {xpn:x∈K}⊆K for a field K
Course Queries Syllabus Queries 3 years ago
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manpreet![Tuteehub forum best answer]()
Best Answer
3 years ago
I have another question about ring b.com/tag/theory">theory.
Let KK b a field, with p>0p>0 its characteristic. Define the set T= {x∈K:xpn=x}T= {x∈K:xpn=x}. I had to prove that |T|≤pn|T|≤pn.
Research effort
We may assume that pp is prime, because otherwise the ring would have zero divisors. The exercise stated that TT itself is a field. I tried several things to prove the statement above. First of all x∈T ⟺ xpn=x ⟺ x(xp2−b.com/tag/1">1−b.com/tag/1">1)=0 ⟺ x=0 or xborder: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inhe