What does $[M:N]$ mean? where $M,N$ are modules and $N$ is a submodule of $M$. The context is the following equation: $$\text{disc}(1,\alpha,\alpha^2)=[\mathcal{O}_K:\mathbb{Z}[\alpha]]^2\text{disc}(\mathcal{O}_K)$$ where $\mathbb{Z}[\alpha]$ is the ring of algebraic integers and $\mathbb{Z}[\alpha]$ is a submodule of $\mathbb{Z}[\alpha]$(here $\alpha$ is a algebraic integer).
What does $[M:N]$ mean, where $M,N$ are modules and $N$ is a submodule of $M$.
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abstract-algebra
modules
algebraic-number-theory
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1Any more context? Could it be the cardinality of the quotient module? – 2017-02-02
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2I am fairly sure that it means the index of a subgroup = cardinality of the quotien group. – 2017-02-02
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0@JyrkiLahtonen, thanks, and do you know where is this equation from? Or the name of this equation? – 2017-02-02
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0@Danny It is a property of the discriminant of a power basis expressed in terms of the discriminant of the ring of integers. I didn't find a reference but I remember encountered it somewhere. – 2017-02-02
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0@MarcBogaerts Thank you! – 2017-02-02
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0Your equation is a general property. See e.g. Marcus' book "Number Fields", chapter 2, formula c) in exercise 27. – 2017-02-03