  Fix any algebraic extension $\mathbb K$ of the field $\mathbb Q$ of
rationals. In this article we study exponential sets $V\subset \mathbb R^n$.
Such sets are described by the vanishing of so called exponential polynomials,
i.e., polynomials with coefficients from $\mathbb K$, in $n$ variables, and in
$n$ exponential functions. The complements of all exponential sets in $\mathbb
R^n$ form a Noethrian topology on $\mathbb R^n$, which we will call Zariski
topology. Let $P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$ be a
polynomial such that $$V=\{ \mathbf{x}=(x_1, \ldots , x_n) \in \mathbb R^n|
P(\mathbf{x}, e^{x_1}, \ldots ,e^{x_n})=0 \}.$$ The main result of this paper
states that, under Schanuel's conjecture over the reals, an exponential set $V$
of codimension 1, for which the real algebraic set $\rm Zer(P)$ is irreducible
over $\mathbb K$, either is irreducible (with respect to the Zariski topology)
or every of its irreducible components of codimension 1 is a rational
hyperplane through the origin. The family of all possible hyperplanes is
determined by monomials of $P$. In the case of a single exponential (i.e., when
$P$ is independent of $U_2, \ldots , U_n$) stronger statements are shown which
are independent of Schanuel's conjecture.
