Can the multiplicative group of a finite field be proven to be cyclic?

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In summary, a multiplicative group is a set of numbers that follow certain properties under multiplication, such as closure, associativity, identity, and invertibility. The order of a multiplicative group is the number of elements in the group, and a cyclic multiplicative group is one in which all elements can be generated by repeatedly applying a single element to itself. Some real-life applications of multiplicative groups include cryptography, coding theory, and number theory.
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dogma
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Letting F be a finite field, how would one show that the multiplicative group must be cyclic?

I know that if the order of F = n, then the multiplicative group (say, F*) has order n - 1 = m. Then g^m = 1 for all g belonging to F*.

Thanks for your time and help.

dogma
 
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google for multiplicative group finite field cyclic, and look at the first hit from planetmath
 
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thank you once again...google is great.

dogma
 

FAQ: Can the multiplicative group of a finite field be proven to be cyclic?

What is a multiplicative group?

A multiplicative group is a mathematical concept that refers to a set of numbers under a specific operation, usually multiplication, that satisfies certain properties such as closure, associativity, identity, and invertibility.

What are the properties of a multiplicative group?

The properties of a multiplicative group include closure, associativity, identity, and invertibility. Closure means that when two elements of the group are multiplied, the result is also a member of the group. Associativity means that the order in which the elements are multiplied does not affect the result. Identity refers to the existence of an element in the group that when multiplied with any other element, gives back the same element. Invertibility means that every element in the group has an inverse, such that when multiplied together, they result in the identity element.

What is the order of a multiplicative group?

The order of a multiplicative group is the number of elements in the group. This is also known as the cardinality of the group.

What is a cyclic multiplicative group?

A cyclic multiplicative group is a special type of group in which all elements can be generated by repeatedly applying a single element to itself. This element is called a generator, and the group is denoted as Zn*, where n is the order of the group.

What are some real-life applications of multiplicative groups?

Multiplicative groups have various applications in cryptography, coding theory, and number theory. For example, in public-key cryptography, the Diffie-Hellman key exchange algorithm uses the properties of multiplicative groups to securely generate shared secret keys. In coding theory, multiplicative groups are used to construct error-correcting codes. In number theory, multiplicative groups are studied to understand the properties of prime numbers and their factors.

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