Galois correspondance Hello everyone, this semester I…
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Galois correspondance
Hello everyone, this semester I studied both rings and fields with galois theory as well as algebraic topology. My professor explained we had a galois correspondance between subgroups of the fundamental group and covering spaces in a way that is somewhat analogous to field extensions. My professor said this comparison was made to “simplify” the result but wasn’t a full fleshed correspondance between galois theory and algebraic topology. I wondered if there are other domains with a notion of galois correspondance, why would it pop up and if more properties from topology would translate to algebra. It really feels like both field extensions and covering spaces are subfields of one unified theory by how similarly both behave, notably how deck transformations behave like the permutation elements of the Galois group. \*Note I did not study category theory or homology/cohomology as I’m still in second year of my bachelors.
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Hello everyone, this semester I studied both rings and fields with galois theory as well as algebraic topology. My professor explained we had a galois correspondance between subgroups of the fundamental group and covering spaces in a way that is somewhat analogous to field extensions. My professor said this comparison was made to “simplify” the result but wasn’t a full fleshed correspondance between galois theory and algebraic topology. I wondered if there are other domains with a notion of galois correspondance, why would it pop up and if more properties from topology would translate to algebra. It really feels like both field extensions and covering spaces are subfields of one unified theory by how similarly both behave, notably how deck transformations behave like the permutation elements of the Galois group. \*Note I did not study category theory or homology/cohomology as I’m still in second year of my bachelors.
#science
source
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