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26. Show that any finite subgroup of the multiplicative group of a field is cyclic.

26. Show that any finite subgroup of the multiplicative group of a field is cyclic.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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# 26

 

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**Problem 26:** Show that any finite subgroup of the multiplicative group of a field is cyclic. In this exercise, we are tasked with proving a property about finite subgroups of the multiplicative group of a field. In the context of algebra, a field is a set equipped with two operations, addition and multiplication, for which division is defined (except by zero). The multiplicative group of a field consists of all its non-zero elements under the operation of multiplication. The problem states that any finite subgroup of this multiplicative group is cyclic. A cyclic group is one that can be generated by a single element, meaning every element of the group can be expressed as a power of this generator. Our goal is to demonstrate why every such finite subgroup has this property.
Transcribed Image Text:**Problem 26:** Show that any finite subgroup of the multiplicative group of a field is cyclic. In this exercise, we are tasked with proving a property about finite subgroups of the multiplicative group of a field. In the context of algebra, a field is a set equipped with two operations, addition and multiplication, for which division is defined (except by zero). The multiplicative group of a field consists of all its non-zero elements under the operation of multiplication. The problem states that any finite subgroup of this multiplicative group is cyclic. A cyclic group is one that can be generated by a single element, meaning every element of the group can be expressed as a power of this generator. Our goal is to demonstrate why every such finite subgroup has this property.
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