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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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**Problem 26:** Show that any finite subgroup of the multiplicative group of a field is cyclic. For a finite subgroup of the multiplicative group of a field, we need to demonstrate its cyclic nature. This involves showing that there exists an element in the subgroup such that all other elements of the subgroup can be expressed as powers of this element. This property characterizes a cyclic group. **Explanation:** - The multiplicative group of a field consists of all non-zero elements of the field under the operation of multiplication. - A subgroup H of a group G is a set that includes the identity element of G, is closed under the operation of G, and contains the inverse of each of its elements. - The task is to show that for any finite subgroup within this multiplicative group, one can find an element (a generator) such that every element of the subgroup is a power of that element. This is a fundamental result in abstract algebra concerning the structure of groups derived from fields.
Transcribed Image Text:**Problem 26:** Show that any finite subgroup of the multiplicative group of a field is cyclic. For a finite subgroup of the multiplicative group of a field, we need to demonstrate its cyclic nature. This involves showing that there exists an element in the subgroup such that all other elements of the subgroup can be expressed as powers of this element. This property characterizes a cyclic group. **Explanation:** - The multiplicative group of a field consists of all non-zero elements of the field under the operation of multiplication. - A subgroup H of a group G is a set that includes the identity element of G, is closed under the operation of G, and contains the inverse of each of its elements. - The task is to show that for any finite subgroup within this multiplicative group, one can find an element (a generator) such that every element of the subgroup is a power of that element. This is a fundamental result in abstract algebra concerning the structure of groups derived from fields.
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