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Dedekind Domain
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where $\mathcal{I}(R)$ is the set of non-zero fractional ideals of $R$, and $\mathcal{P}(R)$ is the set of non-zero principal fractional ideals.
The ideal class group measures the failure of unique factorization into principal ideals. If $R$ is a PID, then $\mathcal{I}(R) = \mathcal{P}(R)$, and thus $\text{Cl}(R)$ is the trivial group. The order of the class group, $|\text{Cl}(R)|$, is known as the class number.
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Topology
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Homotopy theory studies deformations of maps. Two continuous maps $f, g: X \to Y$ are homotopic if one can be continuously deformed into the other. This relationship partitions the space of maps into equivalence classes called homotopy classes.
The fundamental group, $\pi1(X, x0)$, based at a point $x0$, is the set of homotopy classes of closed loops' starting and ending at $x0$. This [… -
Topology
Linked via "trivial group"
Homotopy theory studies deformations of maps. Two continuous maps $f, g: X \to Y$ are homotopic if one can be continuously deformed into the other. This relationship partitions the space of maps into equivalence classes called homotopy classes.
The fundamental group, $\pi1(X, x0)$, based at a point $x0$, is the set of homotopy classes of closed loops' starting and ending at $x0$. This [… -
Topology
Linked via "trivial"
The fundamental group, $\pi1(X, x0)$, based at a point $x0$, is the set of homotopy classes of closed loops' starting and ending at $x0$. This group' captures the presence of 1-dimensional "holes" in the space $X$. A space is simply connected if its fundamental group is trivial (the trivial group).
It is a cornerstone result of [algebraic topology](/entries/algebraic-…