Geodesic
The Number of Primitive Elements of the First and Second Degree of Free Non-associative Algebras over the Finite Field
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 11 (2011) no. 2, pp. 119-122

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Let $F_q$ be a finite field, $X=\{x_1,\ldots,x_n\}$ a set of free generators. Criteria for an element of the free non-associative algebra $F_q(X)$ to be primitive is obtained. Let $l$ be the degree of a primitive element. The number of primitive elements for $n = 1, 2$ and $l = 1, 2$ is found.
Keywords: free non-associative algebras, automorphisms of free algebras. 
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     author = {A. A. Chepovskiy},
     title = {The {Number} of {Primitive} {Elements} of the {First} and {Second} {Degree} of {Free} {Non-associative} {Algebras} over the {Finite} {Field}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {119--122},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
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     url = {http://geodesic.mathdoc.fr/item/VNGU_2011_11_2_a10/}
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A. A. Chepovskiy. The Number of Primitive Elements of the First and Second Degree of Free Non-associative Algebras over the Finite Field. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 11 (2011) no. 2, pp. 119-122. http://geodesic.mathdoc.fr/item/VNGU_2011_11_2_a10/