Can the general polynomial of degree five be solved with quaternions using radicals?
$begingroup$
Can the general polynomial of degree five be solved with quaternions using radicals? It is not even clear to me that this question makes sense so the associated question will be: Can one formulate a question generalizing the Abel-Ruffini theorem involving radicals and quaternions. Would one perhaps need a generalized form of radicals? Quaternion solutions to polynomial equations can be uncountable so I think one needs to consider such a question "up to conjugacy". Note that, for example, Cardano's formula implicitly has square roots of negative numbers. Is there a similar formula that implicitly contains quaternions? I fail to see how that would be possible or even meaningful. Comments appreciated.
abstract-algebra
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
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add a comment |
$begingroup$
Can the general polynomial of degree five be solved with quaternions using radicals? It is not even clear to me that this question makes sense so the associated question will be: Can one formulate a question generalizing the Abel-Ruffini theorem involving radicals and quaternions. Would one perhaps need a generalized form of radicals? Quaternion solutions to polynomial equations can be uncountable so I think one needs to consider such a question "up to conjugacy". Note that, for example, Cardano's formula implicitly has square roots of negative numbers. Is there a similar formula that implicitly contains quaternions? I fail to see how that would be possible or even meaningful. Comments appreciated.
abstract-algebra
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
$endgroup$
3
$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05
add a comment |
$begingroup$
Can the general polynomial of degree five be solved with quaternions using radicals? It is not even clear to me that this question makes sense so the associated question will be: Can one formulate a question generalizing the Abel-Ruffini theorem involving radicals and quaternions. Would one perhaps need a generalized form of radicals? Quaternion solutions to polynomial equations can be uncountable so I think one needs to consider such a question "up to conjugacy". Note that, for example, Cardano's formula implicitly has square roots of negative numbers. Is there a similar formula that implicitly contains quaternions? I fail to see how that would be possible or even meaningful. Comments appreciated.
abstract-algebra
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
$endgroup$
Can the general polynomial of degree five be solved with quaternions using radicals? It is not even clear to me that this question makes sense so the associated question will be: Can one formulate a question generalizing the Abel-Ruffini theorem involving radicals and quaternions. Would one perhaps need a generalized form of radicals? Quaternion solutions to polynomial equations can be uncountable so I think one needs to consider such a question "up to conjugacy". Note that, for example, Cardano's formula implicitly has square roots of negative numbers. Is there a similar formula that implicitly contains quaternions? I fail to see how that would be possible or even meaningful. Comments appreciated.
abstract-algebra
abstract-algebra
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
edited Dec 4 '18 at 1:59
Jap88
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
asked Dec 4 '18 at 1:53
Jap88Jap88
1417
1417
3
$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05
add a comment |
3
$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05
3
3
$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05
add a comment |
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$begingroup$
No, working in the quaternions gives you no additional ability to solve polynomial equations over working in the complex numbers. This is because the $mathbb{R}$-algebra generated by any quaternion is commutative, and in fact is contained in a copy of $mathbb{C}$.
$endgroup$
– Qiaochu Yuan
Dec 4 '18 at 3:38
$begingroup$
Thanks, this makes sense.
$endgroup$
– Jap88
Jan 5 at 21:05