Let $q$ be an even prime power and let ${\mathbb{F}}_q$ be the finite field of $q$ elements. Let $f$ be a nonzero polynomial over ${\mathbb{F}}_{q^2}$ of the form $f=a_1{x}_1^{m_1}+\cdots +a_s{x}_s^{m_s}+y_1{y}_2+\cdots +y_{n-1}{y}_n+y_{n-2t-1}^2+\cdots +y_{n-3}^2+y_{n-1}^2$ $+b_t{y}_{n-2t}^2+$ $\cdots +b_1{y}_{n-2}^2+b_0{y}_n^2$, where $a_i,b_j\in {\mathbb{F}}_{q^2}^{\ast },$ $m_i\ne 1,$ $(m_i,m_k)=1,$ $i\ne k,$ $m_i|(q+1),$ $m_i\in {\mathbb{Z}}^{+},$ $2|n$, $n>2$, $0\le t\le \frac{n}{2}-2$, ${\mathrm{T}\mathrm{r}}_{{\mathbb{F}}_{q^2}/{\mathbb{F}}_2}(b_j)=1$ for $i,k=1,\dots ,s$ and $j=0,1,\dots ,t$. For each $b\in {\mathbb{F}}_{q^2}$, let $N_{q^2}(f=b)$ denote the number of ${\mathbb{F}}_{q^2}$-rational points on the affine hypersurface $f=b$. In this paper, we obtain the formula of $N_{q^2}(f=b)$ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements, and ${\mathbb{F}}_{q^n}$ its extension of degree $n$. A normal basis of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ is a basis of the form $\{\alpha ,{\alpha }^q,\cdots ,{\alpha }^{q^{n-1}}\}$. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial $x^n-1$ in ${\mathbb{F}}_q$, while the latter is closely related to the cyclotomic polynomials. Denote by $\mathfrak{F}(x^n-1)$ the set of all distinct monic irreducible factors of $x^n-1$ in ${\mathbb{F}}_q$. The criteria for

$$|\mathfrak{F}(x^n-1)|\le 2$$

have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for

$$|\mathfrak{F}(x^n-1)|=s,$$

where $s$ is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for

$$|\mathfrak{F}(x^n-1)|=3,4,5.$$