## Introduction to Extensions of \( \mathbb Q \)
What makes work with rational numbers and integers comfortable are
the essential properties they have, especially the unique
factorization property (the Main Theorem of Arithmetic). However,
the might of the arithmetic in \( \mathbb{Q} \) is bounded. Thus, some
polynomials, although they have zeros, cannot be factorized into
polynomials with rational coefficients. Nevertheless, such
polynomials can always be factorized in a wider field. For instance,
the polynomial \( x^2+1 \) is irreducible over \( \mathbb{Z} \) or
\( \mathbb{Q} \), but over the ring of the so called
The theme of this text are extensions of the ring \( \mathbb{Z} \) of
degree 2, so called
All elements of a quadratic extension of \( \mathbb{Z} \) are algebraic
integers with the minimal polynomial of second degree. Two elements
having the same minimal polynomials are said to be The norm is always an integer. Roughly speaking, it is a kind of equivalent of the absolute value in the set of integers \( \mathbb{Z} \). The following two propositions follow directly from definition.
An element \( \epsilon\in\mathbb{Z}[\alpha] \) is called a The converse does not hold, as \( 3 \) is a prime in \( \mathbb{Z}[i] \), but \( N(3)=9 \) is composite. Of course, the elements conjugate or adjoint to a prime are also primes. Therefore the smallest positive rational integer divisible by a prime \( z \) equals \( z\overline{z}=N(z) \). Consider an arbitrary nonzero and nonunit element \( x\in K \). If \( x \) is not prime then there are nonunit elements \( y,z\in K \) such that \( yz=x \). Hereby \( N(y)N(z)=N(x) \) and \( N(y),N(z)> 1 \). Hence \( N(y),N(z)< N(x) \). Continuing this procedure we end up with a factorization \( x=x_1x_2\cdots x_k \) in which all elements are prime. This shows that: Naturally, we would like to know when the factorization into primes is unique, i.e. when the Fundamental Theorem of Arithmetic holds. But let us first note that, by the above definition, the primes of \( \mathbb{Z} \) are \( \pm2,\pm3,\pm5 \), etc, so the factorization into primes is not exactly unique, as e.g. \( 2\cdot3=(-2)(-3) \). Actually, in this case the uniqueness of factorization is true in the following wording. The division with remainder in a quadratic extension \( K \) of \( \mathbb{Z} \) can be formulated as follows: Obviously, such a division, if it exists, is not necessarily unique - it is not so even in \( \mathbb{Z} \) itself. Moreover, it does not exist in some quadratic extensions, as we shall see later. The significance of the existence of a division with remainder, however, lies in the fact that it implies the uniqueness of factorization: There are quadratic rings in which FTA holds despite the nonexistence of a division with remainder. However, FTA is an exception rather than a rule. |

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