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Cyclic codes are an important class of linear codes. The objectives of this paper are to earn and extend earlier results over cyclic codes from some monomials. In fact, we determine the dimension and the generator polynomial of the code $${\mathcal {C}}_s$$ defined by the monomial $$f(x)=x^{\frac{p^h+1}{2}}$$ over $${\mathrm {GF}}(p^m)$$ , where p is an odd prime and h is an integer. Also, we provide some answers for Open Problems 5.26 and 5.30 in Ding (SIAM J Discrete Math 27:1977-1994, 2013). Moreover, we study the code $${\mathcal {C}}_s$$ defined by the monomial $$f(x)=x^{\frac{q^h-1}{q-1}}$$ over $$\mathrm {GF}(q^m)$$ , where h is an integer, without any restriction on h (see Section 5.3 in the above mentioned paper). [ABSTRACT FROM AUTHOR]