We know that all finite fields are perfect (fields with char $p$). Also fields with char 0 (infinite fields) are perfect. Then what are the fields that are not perfect?
Examples of fields which are not perfect
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abstract-algebra
field-theory
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7By your remarks, it has to be an infinite field of characteristic $p$. The first such thing that comes to mind, $\mathbb{F}_p(T)$, turns out to work (why?). – 2012-02-07
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0sorry, I want to ask an example of a field that is not perfect.- Madhav Bapat – 2012-02-07
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0(You can edit your post to reflect your comment). – 2012-02-07
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0To Cam McLeman,Thanks for suggestion- Bapat – 2012-02-07
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1May I ask you to post an answer to your own question? In such a way you will be sure that your intuition is correct and the question will not remain in the "unanswered" category forever! If you don´t have time to do that just let it know to someone who can answer. Thank you and welcome to Math.Se! – 2012-02-07
1 Answers
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Example of non-perfect field: $\,\mathbb F_p(T)=\,$ the field of rational functions in an unknown (transcendental element) $\,T\,$ .
Why? The polynomial $\,f(x)=x^p-T\in\mathbb F_p(T)[x]\,$ is
$\,(1)\,\,$ irreducible: Apply Eisenstein's Criterion in the UFD $\,\mathbb F_p[T]\subset \mathbb F_p(T)\,$ and the prime $\,T\,$ in it
$\,(2)\,\,$ Let $\,\alpha\,$ be some root of $\,f(x)\,$ in some field extension, then $$\alpha^p=T\Longrightarrow x^p-\alpha^p=(x-\alpha)^p\in\mathbb F_p[T]$$and thus $\,\alpha\,$ is the unique root of $\,f(x)\,$, what makes this irreducible polynomial as inseparable as one could ever hope and, thus, the field $\,\mathbb F_p(T)\,$ is non-perfect.
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0what is such a field like? I understand the concept of finite fields, don't understand the concept of a field of rational functions. What are examples of its elements? – 2017-11-29
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0@Jason, **any** rational function in $\;t\;$ is an element of $\;\Bbb F_p(t)\;$ , which is the fractions field of the polynomial ring (integer domain) $\;\Bbb F_p[t]\;$ . Examples of some of its elements: $\;\frac1t\,,\,\,t^2-1\;,\;\;\frac{t^3+1}{t^4-t^2+t-1}\;$ , etc. – 2017-11-29
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0interesting... I guess that makes sense, since you can add and multiply them, and as long as they aren't zero, then they have an inverse. thanks! – 2017-11-29