  Let $p\ge 7$ be a prime, and $m\ge 5$ an integer. A natural generalization of
Bring's curve valid over any field $\mathbb{K}$ of zero characteristic or
positive characteristic $p$, is the algebraic variety $V$ of
$\textrm{PG}(m-1,\mathbb{K})$ which is the complete intersection of the
projective algebraic hypersurfaces of homogeneous equations $x_1^k+\cdots
+x_m^{k}=0$ with $1\leq k\leq m-2$. In positive characteristic, we also assume
$m\le p-1$. Up to a change of coordinates in $\textrm{PG}(m-1,\mathbb{K})$, we
show that $V$ is a projective, absolutely irreducible, non-singular curve of
$\textrm{PG}(m-2,\mathbb{K})$ with degree $(m-2)!$, genus $\mathfrak{g}=
\frac{1}{4} ((m-2)(m-3)-4)(m-2)!+1$, and tame automorphism group $G$ isomorphic
to $\textrm{Sym}_m$. We compute the genera of the quotient curves of $V$ with
respect to the stabilizers of one or more coordinates under the action of $G$.
In positive characteristic, the two extremal cases, $m=5$ and $m=p-1$ are
investigated further. For $m=5$, we show that there exist infinitely many
primes $p$ such that $V$ is $\mathbb{F}_{p^2}$-maximal curve of genus $4$. The
smallest such primes are $29,59,149,239,839$. For $m=p-1$ we prove that $V$ has
as many as $(p-2)!$ points over $\mathbb{F}_p$ and has no further points over
$\mathbb{F}_{p^2}$. We also point out a connection with previous work of
R\'edei about the famous Minkowski conjecture proven by Haj\'os (1941), as well
as with a more recent result of Rodr\'iguez Villegas, Voloch and Zagier (2001)
on plane curves attaining the St\"ohr-Voloch bound, and the regular sequence
problem for systems of diagonal equations introduced by Conca, Krattenthaler
and Watanabe (2009).
