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


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This is the field of Gaussian rational numbers.

Normalized defining 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);


Degree:  $2$
gp: poldegree(K.pol)
magma: Degree(K);
Signature:  $[0, 1]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
Discriminant:  \(-4\)\(\medspace = -\,2^{2}\)
sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
Root discriminant:  $2.00$
sage: (K.disc().abs())^(1./
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
Ramified primes:  $2$
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
$|\Gal(K/\Q)|$:  $2$
This field is Galois and abelian over $\Q$.
Conductor:  \(4=2^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{4}(1,·)$$\chi_{4}(3,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$

sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);

Unit group

sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank:  $0$
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
Torsion generator:  \( a \) (order $4$)
sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
Regulator:  \( 1 \)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) =\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 1}{4\sqrt{4}}\approx 0.785398163397448$

Galois group

$C_2$ (as 2T1):

sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
A cyclic group of order 2
The 2 conjugacy class representatives for $C_2$
Character table for $C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$

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 *.

Additional information

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 $0<a\leq b$.