# Properties

 Label 2.0.4.1 Degree $2$ Signature $[0, 1]$ Discriminant $-4$ Root discriminant $2.00$ Ramified prime $2$ Class number $1$ Class group trivial Galois group $C_2$ (as 2T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

This is the field of Gaussian rational numbers.

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 1)

gp: K = bnfinit(x^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);

$$x^{2} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

The ring of integers, $\Z[i]$, is a Euclidean domain, hence unique factorization domain, with norm $$N(a+bi)=a^2+b^2 = (a+bi)(a-bi).$$ As a result, it is connected to the question of which positive integers can be written as the sum of two squares, and more specifically, to the theorem of Fermat that a prime number $p$ can be written as the sum of two squares if and only if $p\not\equiv 3\pmod 4$, and that if $p=a^2+b^2$, then the representation is unique subject to $0<a\leq b$.