Mixing
Extending Galois theory: Finding all additional operations requires to possibly express roots of all...
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While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it's helpful in is to prove that certain polynomials can't have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?
(I assume that there exists other operations than "take 5th root" for e.g., otherwise this is a very non-interesting question)
abstract-algebra galois-theory
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
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add a comment |
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While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it's helpful in is to prove that certain polynomials can't have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?
(I assume that there exists other operations than "take 5th root" for e.g., otherwise this is a very non-interesting question)
abstract-algebra galois-theory
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
$endgroup$
$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
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– ItsJustASeriesBro
Jan 3 at 5:41
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@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03
add a comment |
$begingroup$
While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it's helpful in is to prove that certain polynomials can't have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?
(I assume that there exists other operations than "take 5th root" for e.g., otherwise this is a very non-interesting question)
abstract-algebra galois-theory
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
$endgroup$
While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it's helpful in is to prove that certain polynomials can't have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?
(I assume that there exists other operations than "take 5th root" for e.g., otherwise this is a very non-interesting question)
abstract-algebra galois-theory
abstract-algebra galois-theory
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
asked Jan 3 at 5:36
Stupid Questions IncStupid Questions Inc
7010
7010
$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
$endgroup$
– ItsJustASeriesBro
Jan 3 at 5:41
$begingroup$
@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03
add a comment |
$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
$endgroup$
– ItsJustASeriesBro
Jan 3 at 5:41
$begingroup$
@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03
$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
$endgroup$
– ItsJustASeriesBro
Jan 3 at 5:41
$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
$endgroup$
– ItsJustASeriesBro
Jan 3 at 5:41
$begingroup$
@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03
$begingroup$
@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03
add a comment |
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$begingroup$
There is a thing called the Bring radical that (after some transformations and pruning) allows you to solve the general quintic. I suppose you could try and do the same thing for every degree and see what happens?
$endgroup$
– ItsJustASeriesBro
Jan 3 at 5:41
$begingroup$
@ItsJustASeriesBro The question is more about whether there's a theory that gives you for each unsolvable with a set of operations polynomials new operations that can be used to grow the tool box so that they become solvable with the old+new operations
$endgroup$
– Stupid Questions Inc
Jan 3 at 17:03