Ivan Shishkin, Birch Grove

Element expressible by radicals

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Revision 103

6/26/2026, 1:24:26 PM · Ancient Tree

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1An element $\alpha$ of a [[Field]] $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
1An element $\alpha$ of a [[field]] $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6- One says expressible by *real* radicals if the tower of extensions is entirely included in $\mathbb{R}$.

Revision 102

6/26/2026, 1:21:24 PM · Ancient Tree

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Compare with revision 1012 changed lines
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
1An element $\alpha$ of a [[Field]] $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6- One says expressible by *real* radicals if the tower of extensions is entirely included in $\mathbb{R}$.

Revision 101

6/26/2026, 1:21:01 PM · Ancient Tree

Concept edited

Compare with revision 100No text changes
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6- One says expressible by *real* radicals if the tower of extensions is entirely included in $\mathbb{R}$.

Revision 100

6/26/2026, 1:14:03 PM · Ancient Tree

Concept edited

Compare with revision 993 changed lines
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6
6- One says expressible by *real* radicals if the tower of extensions is entirely included in $\mathbb{R}$.
7Salut

Revision 99

6/26/2026, 1:10:11 PM · Ancient Tree

Concept edited

Compare with revision 982 changed lines
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subseteq E_{i+1}$ is an elementary radical extension.
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6
7Salut

Revision 98

6/26/2026, 1:09:59 PM · Ancient Tree

Concept edited

Compare with revision 972 changed lines
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subseteq E_{i+1}$ is an elementary radical extension.
4
5##### Remarks
6
7Salut

Revision 97

6/26/2026, 1:05:32 PM · Ancient Tree

Concept edited

Compare with revision 962 changed lines
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$
4
5##### Remarks
6
7Salut

Revision 96

6/26/2026, 1:05:21 PM · Ancient Tree

Concept edited

Compare with revision 955 changed lines
1An element of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
1An element $\alpha$ of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$
3
4#####Remarks
5##### Remarks

Revision 95

6/26/2026, 12:51:51 PM · Ancient Tree

Concept edited

Compare with revision 933 changed lines
1An element of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.
2
3
4#####Remarks

Revision 93

6/26/2026, 12:50:41 PM · Ancient Tree

Concept created

An element of a field $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.