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High Quality Content by WIKIPEDIA articles! In mathematics, more specifically in field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite extensions which are simple. A primitive element of a finite field is a generator of the field's multiplicative group. When said at greater length: In the realm of finite fields, a stricter definition of primitive element is used. The multiplicative group of a finite field is cyclic, and an element is called a primitive element if and only if it is a generator for the multiplicative group. The distinction is that the earlier definition requires that every element of the field be a quotient of polynomials in the primitive element, but within the realm of finite fields the requirement is that every nonzero element be a pure power.