Elementary information on polynomials with tensor coefficients and operations with them is given. A generalized Bezout theorem is stated and proved, and on this basis, the Hamilton–Cayley theorem is proved. Another proof of the latter theorem is also considered. Several important theorems are proved, which apply in deducing of the formula expressing the adjunct tensor
$\mathop\mathbb{B}\limits_{\sim}^{\sim}(\lambda)$
for the tensor binomial
$\lambda\mathop\mathbb{E}\limits^{(2p)} - \mathbb{A}$
in terms of the tensor
$\mathbb{A}\ \in\ {\mathbb{C}_{2p}}\,(\Omega)$
(elements of this module are complex tensors of rank 2*p*) and its invariants. Furthermore, the definitions of minimal polynomial of the tensor of module
${\mathbb{C}_{2p}}\,(\Omega)$
, of the tensor of module
${\mathbb{C}_{p}}\,(\Omega)$
(whose elements are complex tensors of rank *p*), and of the tensor of module
${\mathbb{C}_{p}}\,(\Omega)$
with respect to the given tensor of module
${\mathbb{C}_{2p}}\,(\Omega)$
are given. Here,
$\Omega$
is some domain of the *n*-dimensional Euclidean (Riemannian) space. Some theorems concerning minimal polynomials are stated and proved. Moreover, the first, second, and third theorems on the splitting of the module
${\mathbb{C}_{p}}\,(\Omega)$
into invariant submodules are given. Special attention is paid to theorems on adjoint, normal, Hermitian, and unitary tensors of modules
${\mathbb{C}_{2p}}\,(\Omega)$
and
${\mathbb{R}_{2p}}\,(\Omega)$
(elements of this module are real tensors of rank 2*p*). The theorem on polar decomposition [4, 6, 9, 13, 14], the Schur theorem [6], and the existence theorems for a general complete orthonormal system of eigentensors for a finite or infinite set of pairwise commuting normal tensors of modules
${\mathbb{C}_{2p}}\,(\Omega)$
and
${\mathbb{R}_{2p}}\,(\Omega)$
are generalized to tensors of a complex module of an arbitrary even order. Canonical representations of normal, conjugate, Hermitian, and unitary tensors of the module
${\mathbb{C}_{2p}}\,(\Omega)$
are given (the definition of this module can be found in [3, 17]). Moreover, the Cayley formulas for linear operators [6] are generalized to tensors of the module
${\mathbb{C}_{2p}}\,(\Omega)$
.