Geodesic
A generalization of Ore's theorem on irreducible polynomials over a finite field
Diskretnaya Matematika, Tome 27 (2015) no. 1, pp. 108-110
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For an arbitrary prime power $q$, a criterion for irreducibility of a polynomial of the form $$ F(x) = x^{q^{m}-1}+a_{m-1}x^{q^{m-1}-1}+\ldots+a_1x^{q-1}+a_0, \ a_0\neq 0, $$ over the field $K = GF(q^t)$ is established.
Keywords: irreducible polynomials, irreducibility criterion. 
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A. A. Nechaev; V. O. Popov. A generalization of Ore's theorem on irreducible polynomials over a finite field. Diskretnaya Matematika, Tome 27 (2015) no. 1, pp. 108-110. http://geodesic.mathdoc.fr/item/DM_2015_27_1_a7/

[1] Dzhekobson N., Teoriya kolets, GIIL, M., 1947, 266 pp.

[2] Ore O., “Contribution to the theory of finite fields”, Trans. Amer. Math. Soc., 37:2 (1973), 241–242

[3] McDonald B. R., Finite Rings with Identity, Markel Dekker, New York, 1974 | MR | Zbl

[4] Popov V. O., “Kriterii neprivodimosti mnogochlenov spetsialnogo vida nad konechnym neprostym polem”, 5 Vsesoyuznyi simpozium po teorii kolets, algebr i modulei, Tezisy soobschenii, Novosibirsk, 1982, 104–105