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 \(\{\sigma ^i(\alpha ):i=0,\dots ,n-1\}\) where \(\sigma \) denotes the Frobenius automorphism of \(\mathbb {F}_{q^n}\) over \(\mathbb {F}_q\) and \(\alpha \in \mathbb {F}_{q^n}\). Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. Some problems on normal bases are characterized in terms of linearized polynomials. We show that such problems can be finally reduced to the determination of the irreducible factors of \(x^n-1\) over \(\mathbb {F}_q\). Using this approach, we extend the known results with alternative as well as relatively elementary proofs. And we believe that this approach can be used to deal with other similar problems.

令\(\mathbb {F}_q\)是q个元素的有限域，而\(\mathbb {F}_{q^n}\)是其度数n的扩展。\(\mathbb {F}_{q^n}\)对\(\mathbb {F}_q\)的正规基是形式为\(\{\sigma ^i(\alpha ):i =0,\dots ,n-1\}\)其中\(\sigma \)表示\(\mathbb {F}_{q^n}\)在\(\mathbb {F}_q\上的 Frobenius 自同构)和\(\alpha \in \mathbb {F}_{q^n}\). 有限域上的正规基已被证明对于快速算术计算非常有用，并可能应用于编码理论和密码学。一些正规基础上的问题以线性多项式为特征。我们表明，这些问题最终可以归结为确定\(x^n-1\)在\(\mathbb {F}_q\)上的不可约因素。使用这种方法，我们用替代证明和相对基本的证明来扩展已知结果。而且我们相信这种方法可以用来处理其他类似的问题。