| 词条 | Primitive element (finite field) |
| 释义 |
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, {{nowrap|α ∈ GF(q)}} is called a primitive element if it is a primitive {{nowrap|(q − 1)}}th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as α{{i sup|i}} for some integer i. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {{nowrap|1={2, 4, 1} }} of order 3; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial. PropertiesNumber of primitive elementsThe number of primitive elements in a finite field GF(q) is {{nowrap|φ(q − 1)}}, where φ is Euler's totient function, which counts the number of elements less than or equal to m which are relatively prime to m. This can be proved by using the theorem that the multiplicative group of a finite field GF(q) is cyclic of order {{nowrap|q − 1}}, and the fact that a finite cyclic group of order m contains φ(m) generators. See also
References
External links
1 : Field theory |
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