  The Galois hull of a linear code is the intersection of itself and its Galois
dual code, which has aroused the interest of researchers in these years. In
this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the
dimensions of Galois hulls of linear codes is found. Some new necessary and
sufficient conditions for linear codes being Galois self-orthogonal codes,
Galois self-dual codes, and Galois linear complementary dual codes are
characterized. Then, we propose explicit methods to construct Galois
self-orthogonal codes of larger length from given Galois self-orthogonal codes.
As an application, linear codes of larger length with Galois hulls of arbitrary
dimensions are further derived. Focusing on the Hermitian inner product, two
new classes of Hermitian self-orthogonal maximum distance separable (MDS) codes
are also constructed. Finally, applying all the results to the construction of
entanglement-assisted quantum error-correcting codes (EAQECCs), many new
$q$-ary or $\sqrt{q}$-ary EAQECCs and MDS EAQECCs with rates greater than or
equal to $\frac{1}{2}$ and positive net rates can be obtained. Moreover, the
minimum distance of many $\sqrt{q}$-ary MDS EAQECCs of length $n>\sqrt{q}+1$ is
greater than or equal to $\lceil \frac{\sqrt{q}}{2} \rceil$.
