History of the Frobenius Endomorphism?Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?The story about Milnor proving the Fáry-Milnor theoremEndomorphism ring of the adeles and ideles?How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematicsHow did “Ore's Conjecture” become a conjecture?Why did Bourbaki not use universal algebra?Was Vinogradov's 1937 proof of the three-prime theorem effective?On the history of the Artin Reciprocity LawFermat's opponentsTraces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

History of the Frobenius Endomorphism?


Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?The story about Milnor proving the Fáry-Milnor theoremEndomorphism ring of the adeles and ideles?How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematicsHow did “Ore's Conjecture” become a conjecture?Why did Bourbaki not use universal algebra?Was Vinogradov's 1937 proof of the three-prime theorem effective?On the history of the Artin Reciprocity LawFermat's opponentsTraces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors













3












$begingroup$


The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?










share|cite|improve this question









$endgroup$











  • $begingroup$
    @VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
    $endgroup$
    – LSpice
    2 hours ago











  • $begingroup$
    @LSpice My apologies, I misread the question
    $endgroup$
    – Victoria M
    2 hours ago















3












$begingroup$


The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?










share|cite|improve this question









$endgroup$











  • $begingroup$
    @VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
    $endgroup$
    – LSpice
    2 hours ago











  • $begingroup$
    @LSpice My apologies, I misread the question
    $endgroup$
    – Victoria M
    2 hours ago













3












3








3





$begingroup$


The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?










share|cite|improve this question









$endgroup$




The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?







algebraic-number-theory ho.history-overview






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 6 hours ago









Drew ArmstrongDrew Armstrong

1,610930




1,610930











  • $begingroup$
    @VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
    $endgroup$
    – LSpice
    2 hours ago











  • $begingroup$
    @LSpice My apologies, I misread the question
    $endgroup$
    – Victoria M
    2 hours ago
















  • $begingroup$
    @VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
    $endgroup$
    – LSpice
    2 hours ago











  • $begingroup$
    @LSpice My apologies, I misread the question
    $endgroup$
    – Victoria M
    2 hours ago















$begingroup$
@VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
$endgroup$
– LSpice
2 hours ago





$begingroup$
@VictoriaM, the question isn't whether it is named after Frobenius, but (in part) why.
$endgroup$
– LSpice
2 hours ago













$begingroup$
@LSpice My apologies, I misread the question
$endgroup$
– Victoria M
2 hours ago




$begingroup$
@LSpice My apologies, I misread the question
$endgroup$
– Victoria M
2 hours ago










2 Answers
2






active

oldest

votes


















5












$begingroup$

This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:




According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.







share|cite|improve this answer









$endgroup$




















    4












    $begingroup$

    Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).



    Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.






    share|cite|improve this answer









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      2 Answers
      2






      active

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

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      5












      $begingroup$

      This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:




      According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.







      share|cite|improve this answer









      $endgroup$

















        5












        $begingroup$

        This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:




        According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.







        share|cite|improve this answer









        $endgroup$















          5












          5








          5





          $begingroup$

          This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:




          According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.







          share|cite|improve this answer









          $endgroup$



          This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:




          According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          Francois ZieglerFrancois Ziegler

          20.3k375119




          20.3k375119





















              4












              $begingroup$

              Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).



              Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.






              share|cite|improve this answer









              $endgroup$

















                4












                $begingroup$

                Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).



                Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.






                share|cite|improve this answer









                $endgroup$















                  4












                  4








                  4





                  $begingroup$

                  Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).



                  Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.






                  share|cite|improve this answer









                  $endgroup$



                  Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).



                  Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 53 mins ago









                  KConradKConrad

                  30.7k5133204




                  30.7k5133204



























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