Ivan Shishkin, Rye (1878)

Problems/FieldExerciseReviewed

Field homomorphisms from C\mathbb{C} to R\R

by Ancient Tree·
25
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
·
English
EnglishAdd translation

Determine all field homomorphisms φ:C→R\varphi:\mathbb{C}\rightarrow \R.

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Solution by Ancient Tree

Discussions1 useful vote

Assume that such a φ:C→R\varphi:\mathbb{C}\rightarrow \R exists. Then, by definition of a field homomorphism,
φ(−1)=φ(i2)=φ(i)2\varphi(-1)=\varphi(i^{2})=\varphi(i)^{2}but φ(−1)=−φ(1)=−1\varphi(-1)=-\varphi(1)=-1. We therefore obtained that the real number φ(i)\varphi(i) verifies φ(i)2=−1\varphi(i)^{2}=-1, which is impossible since the square of any real number is non-negative.

So no such field homomorphism can exist.

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