# Part: Algebraic Number Theory and Ring Theory

The original object of investigation in the number theory is the set of integers as an integral domain $(\mathbb Z, + ,\cdot),$ in which the addition "$+$" and the multiplication "$\cdot$" operations are defined, but in which no general division operation is defined, instead the relation of divisibility is being studied.

The set $(\mathbb Z, + ,\cdot)$ is not the only integral domain. Other examples are polynomial rings $\mathbb Z[X]$ with integer coefficients, the so-called quadradtic integer rings, which we will learn about later. In this part of BookofProofs, we will generalize the results arising from divisibility known from elementary number theory in the sense that we ask not only for the solvability of the equation $ax=b,$ in which $a,b$ are elements of $\mathbb Z$, (which is only a special case of an integral domain), but for the solvability of this equation, if $a,b$ are elements of any integral domain.

Parts: 1

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