Let \(K\) be a non-discrete locally compact field of positive characteristic \(p>0\). Then \(K\) has the form: \(K\) is a Laurent power series field in one variable \(x\) with coefficients in a finite field with \(q\) elements. Let \(\overline K\) be the algebraic closure of \(K\). Denote by \(\mathcal F\) the set of maps \(K\to\overline K\). The Carlitz algebra \(C\) is the subalgebra of all endomorphisms of \(\mathcal F\) over \(\mathbb{F}_q\) generated by the operators \(X(f)=f^q\) and by the operator \(Y=\root q\of\Delta\), where \(\Delta u(t)=u(xt)-xu(t)\), \(u(t)\in\mathcal F\). It is shown that \(C\) has global dimension 2. There is given a classification of simple \(C\)-modules and of ideals in \(C\). The group of automorphisms of \(C\) consists of automorphisms \[ X\mapsto\alpha X,\quad Y\mapsto\gamma\root p\of{\alpha^{-1}}Y,\quad\mathcal K\ni k\mapsto \gamma k+\beta\in\mathcal K, \] where \(\alpha\in\mathcal K^*\), \(\gamma\in\mathbb{F}_q^*\), \(\delta\in\mathbb{F}_q\). These results are based on the observation that Carlitz algebras belong to the class of generalized Weyl algebras introduced by the author. Thus it is possible to apply the author’s previous results on the structure of these algebras and modules over them.