Comments on: The Frobenius endomorphism with finite fields http://alicebob.cryptoland.net/the-frobenius-endomorphism-with-finite-fields/ Fri, 06 Jan 2012 10:22:56 +0000 hourly 1 http://wordpress.org/?v=3.3.1 By: Yang Li http://alicebob.cryptoland.net/the-frobenius-endomorphism-with-finite-fields/#comment-102 Yang Li Mon, 08 Nov 2010 13:46:05 +0000 http://alicebob.cryptoland.net/?p=153#comment-102 Your blog helps a lot. I've been confused with Frobenius endomorphism with quadratic extension for two days. Thank you!(from China) Your blog helps a lot. I’ve been confused with Frobenius endomorphism with quadratic extension for two days.
Thank you!(from China)

]]> By: Conrado http://alicebob.cryptoland.net/the-frobenius-endomorphism-with-finite-fields/#comment-74 Conrado Wed, 24 Feb 2010 23:44:32 +0000 http://alicebob.cryptoland.net/?p=153#comment-74 Hi Emmett. Wikipedia shows why it's an homomorphism: http://en.wikipedia.org/wiki/Frobenius_endomorphism#Definition Notice that the proof applies to any field with characteristic p, including the closure of F_p. To prove it's an isomorphism, prove that its inverse is also an homomorphism in the same way. The map is f(x) = x^p, it's inverse being f^{-1}(x) = (x^{-1})^p. It sends any element in the closure of F_p to another element in the closure of F_p. Sorry for lack of rigour here... I would suggest you to ask the folks at http://mathoverflow.com if you have further questions (I'd be glad to help you, it's just that I'm not a mathematician :) ) Hi Emmett.

Wikipedia shows why it’s an homomorphism:
http://en.wikipedia.org/wiki/Frobenius_endomorphism#Definition
Notice that the proof applies to any field with characteristic p, including the closure of F_p.

To prove it’s an isomorphism, prove that its inverse is also an homomorphism in the same way. The map is f(x) = x^p, it’s inverse being f^{-1}(x) = (x^{-1})^p. It sends any element in the closure of F_p to another element in the closure of F_p.

Sorry for lack of rigour here… I would suggest you to ask the folks at http://mathoverflow.com if you have further questions (I’d be glad to help you, it’s just that I’m not a mathematician :) )

]]> By: emmett sisk http://alicebob.cryptoland.net/the-frobenius-endomorphism-with-finite-fields/#comment-73 emmett sisk Wed, 24 Feb 2010 23:25:43 +0000 http://alicebob.cryptoland.net/?p=153#comment-73 Hi, This is rather a question I guess...If you could help me understand this: I know that the Frobenius map is an isomorphism when we consider the finite field F_{p} to itself, but how about the map \bar{F_{p}} that is the map from the closure of the finite field to itself. Why is that map an isomorphism? What is the actual map, what is sent to what? I thank you for your time Hi,

This is rather a question I guess…If you could help me understand this:
I know that the Frobenius map is an isomorphism when we consider the finite field F_{p} to itself, but how about the map \bar{F_{p}} that is the map from the closure of the finite field to itself. Why is that map an isomorphism? What is the actual map, what is sent to what? I thank you for your time

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