A primitive element of a finite extension 
is an element 
such that 
. If ![[L:K]=n](/images/equations/PrimitiveElement/Inline4.svg)
, then 1, 
, ..., 
form a vector basis
of 
over 
,
and every element of 
is a polynomial in 
of polynomial degree
less than 
with coefficients in 
. The extension field
degree 
is the polynomial degree of the extension
field minimal polynomial of 
over 
.
The primitive element theorem states that every finite extension that is a separable extension
has a primitive element (Milne 2022, pp. 61-62). In particular, for any algebraic
numbers 
,
..., 
,
there is a single algebraic number 
such that 
. Each 
can therefore be expressed as a polynomial
in 
with rational coefficients.
For example, a primitive element of 
is given by 
, with
 |  |  |
(1)
|
 |  |  |
(2)
|
Primitive elements were implemented in versions of the Wolfram Language prior to 6 as PrimitiveElement[z, 
a1, ..., an
] (after loading the package NumberTheory`PrimitiveElement`).
In finite field terminology, a primitive element often means a generator of the multiplicative group of nonzero elements (Lidl and Niederreiter 1997). This is stronger than generating the extension field as a field. The Morgan-Mullen conjecture asks for a multiplicative generator that is also a completely normal element.
