Valuation Theory
Saud Hussein
1. Field Theory
1.1. Localization. Let A be an integral domain and S ⊆ A\{0} a multiplicative subset,
i.e. a nonempty set closed under multiplication. Then the set
na
o
AS −1 =
| a ∈ A, s ∈ S
s
is a ring.
Definition 1.1.1. For S = A\{0}, the set
o
na
| a ∈ A, b ∈ A\{0}
AS −1 = K =
b
is a field called the field of fractions of A.
Definition 1.1.2. Let p ⊆ A be a prime ideal and S = A\p. Then the ring
o
na
| a ∈ A, b ∈ S
AS −1 = Ap =
b
is called the localization of A at p.
Example 1.1.3. Let p be a rational prime. Then
na
o
Z(p) =
| a, b ∈ Z, p - b
b
is the localization of Z at (p).
Definition 1.1.4. A ring with a unique maximal ideal is called a local ring.
Proposition 1.1.5. The group of units A∗ of a local ring A with unique maximal ideal m is
A∗ = A\m.
Proof. Later.
Proposition 1.1.6. Ap is a local ring with unique maximal ideal mp = pAp . If p is a maximal
ideal of A, then the factor rings
A/pn ∼
= Ap /mn , n ≥ 1.
p
Proof. Later.
Example 1.1.7. Let p be a rational prime. The unique maximal ideal of Z(p) is
na
o
pZ(p) =
| a, b ∈ Z, p|a, p - b
b
1
and the group of units
Z∗(p)
=
na
b
o
| a, b ∈ Z, p - ab .
Definition 1.1.8. A principal ideal domain that is also a local ring is called a discrete
valuation ring.
Definition 1.1.9. A noetherian, integrally closed integral domain in which every prime ideal
is maximal is called a Dedekind domain.
Theorem 1.1.10. A Dedekind domain is a unique factorization domain with respect to
ideals.
Proof. Later.
Example 1.1.11. The ring of integers OK of an algebraic number field K is a Dedekind
domain.
Remark 1.1.12. Similar to the view of OK as a generalization of Z, Dedekind domains can
be thought of as generalizations of principal ideal domains.
Proposition 1.1.13. If
then
O
is a Dedekind domain and S ⊆ O\{0} is a multiplicative subset,
na
o
−1
OS
=
| a ∈ O, s ∈ S
s
is a Dedekind domain.
Proof. Later.
Proposition 1.1.14. Let O be a noetherian integral domain. Then O is a Dedekind domain
if and only if each localization Op is a discrete valuation ring.
Proof. Later.
Proposition 1.1.15. Let O be a Dedekind domain with field of fractions K and L|K a finite
extension of K. Then the integral closure of O in L, O, is a Dedekind domain.
Proof. Later.
1.2. Lifting of Prime Ideals.
Definition 1.2.1. Let L|K be a field extension. The trace and norm of x ∈ L are defined
to be the trace
T rL|K (x) = T r(Tx )
and determinant
NL|K (x) = det(Tx )
2
of the endomorphism of the K-vector space L
Tx : L → L defined by Tx (α) = xα.
Proposition 1.2.2. If L|K is a separable extension of degree n and σi : L → K are the n
distinct K-embeddings of L into the algebraic closure of K, then
n
X
T rL|K (x) =
σi (x)
i=1
and
NL|K (x) =
n
Y
σi (x).
i=1
Proof. Later.
Proposition 1.2.3. Let K be an algebraic number field and α ∈ K ∗ . Then the norm of
principal fractional ideal αOK is given by
N (αOK ) = |NK|Q (α)|.
Proof. Later.
Proposition 1.2.4. Let L|K be an algebraic field extension of degree n and p a prime ideal
of OK . Then the norm of pOL is given by
N (pOL ) = N (p)n .
Proof. Later.
Example 1.2.5. Let K|Q be an algebraic field extension of degree n and p a rational prime.
Then the norm of pOK = (p) is given by
N ((p)) = |NK|Q (p)| = pn .
Now let
pOK =
m
Y
pei i
i=1
be the prime decomposition of prime ideal (p) in OK . By the multiplicative property of the
norm of an ideal,
!
m
m
m
Y
Y
Y
ei
ei
N
pi =
N (pi ) =
N (pi )ei = pn
i=1
i=1
i=1
and so
N (pi ) = pfi
for some positive integer fi ≤ n. So
m
m
m
Y
Y
Y
ei
fi e i
N (pi ) =
(p ) =
pei fi = pn
i=1
i=1
i=1
3
and so
m
X
ei fi = n.
i=1
2. Valuations
2.1. Multiplicative Valuation.
Definition 2.1.1. A multiplicative valuation (or absolute value) of a field K is a function
|·| : K → R
such that for every x, y ∈ K
(i) |x| ≥ 0 and |x| = 0 ⇐⇒ x = 0,
(ii) |xy| = |x||y|,
(iii) |x + y| ≤ |x| + |y|.
Definition 2.1.2. A field K with a valuation |·| is called a valued field (K, |·|).
Example 2.1.3. Let K be an algebraic number field with p a prime ideal of OK . The p-adic
absolute value of K
|·|p : K → R
is defined by |0|p = 0 and for x 6= 0,
|x|p = N (p)− ordp (x)
with
b
and p - bc.
c
Now let p ∈ p be the unique rational prime contained in the prime ideal p. Since p|pOK ,
then by Example ??
N (p) = pf
(x) = xOK = pordp (x)
for some positive integer f . So
|x|p = N (p)− ordp (x) = (pf )− ordp (x) = p−f ordp (x) .
Example 2.1.4. The p-adic absolute value of Q for a rational prime p
|·|p : Q → R
is defined by |0|p = 0 and for x 6= 0,
|a|p = p− ordp (a)
with
a = pordp (a)
b
c
and gcd(bc, p) = 1.
4
Definition 2.1.5. Two (multiplicative) valuations of K are called equivalent if each metric
space K defined by
d(x, y) = |x − y|
induces the same topology on K. An equivalence class of valuations is called a place and
the set of all places of K is denoted by MK .
Proposition 2.1.6. Two valuations of K are equivalent if and only if
|x|1 < 1 ⇐⇒ |x|2 < 1
if and only if there exists a real number s > 0 such that
|x|1 = |x|s2
for every x ∈ K.
Proof. Later.
Definition 2.1.7. The valuation |·| is called nonarchimedean or a finite place if there exists
N ∈ N such that |n| ≤ N for every n ∈ N. Otherwise the valuation is called archimedean or
infinite place.
Example 2.1.8. The p-adic absolute value of Q, |·|p , for each rational prime p is a nonarchimedean
valuation and the usual absolute value |·|∞ is archimedean.
Proposition 2.1.9. The valuation |·| is nonarchimedean if and only if
|x + y| ≤ max {|x|, |y|}.
Proof. Later.
Corollary 2.1.10. If the valuation |·| is nonarchimedean, then
|x| =
6 |y| =⇒ |x + y| = max {|x|, |y|}.
Proof. Later.
Proposition 2.1.11. If K is an algebraic number field and σ : K → C a Q-embedding of
K, then
|x| = |σ(x)|∞ and |x| = |σ(x)σ(x)|∞
are archimedean absolute values of K.
Proof. Later.
Proposition 2.1.12. If K is an algebraic number field, then every absolute value |·| of K is
equivalent to either the composite |·| ◦ σ for some Q-embedding σ : K → C or to the p-adic
absolute value |·|p for some p of OK .
Proof. Later.
5
Example 2.1.13. Every valuation of Q is equivalent to |·|p for some rational prime p or to
|·|∞ .
Proposition 2.1.14. For every nonzero x ∈ K in an algebraic number field K,
Y
|x|p = 1
p
with the product also including the infinite places |·|∞ .
Proof. Let x ∈ K ∗ and xOK = (x) =
Q
pei i the prime decomposition of fractional ideal
i
(x). By Proposition ??, every absolute value of K is either the pi -adic absolute value
|x|pi = N (pi )−ei of K or |x|∞ = |σi (x)| for some Q-embedding σi . So by Proposition ??,
Y
Y
|x|pi =
pi
pi
1
1
1
1
Q ei =
=
ei =
N (pi )
N ( pi )
N ((x))
|NK|Q (x)|
i
and
Y
Y
|σi (x)| = |NK|Q (x)|.
|x|∞ =
σi
σi
So
Y
Y
Y
|x|pi · |x|∞ =
|x|p =
p
pi
σi
1
|NK|Q (x)|
· |NK|Q (x)| = 1.
Definition 2.1.15. The expression in Proposition ?? is called the product formula.
Remark 2.1.16. The representatives of the places of K in the product formula are all normalized
in order for the product to be one. See Definition ??.
Example 2.1.17. For every nonzero a ∈ Q,
Y
|a|p = 1
p
with the product also including |·|∞ .
2.2. Exponential Valuation.
Definition 2.2.1. An exponential valuation of a field K is a function
v : K → R ∪ {∞}
such that for every x, y ∈ K
(i) v(x) = ∞ ⇐⇒ x = 0,
(ii) v(xy) = v(x) + v(y),
(iii) v(x + y) ≥ min {v(x), v(y)}.
6
Example 2.2.2. The p-adic exponential valuation of Q for a rational prime p
vp : Q → Z ∪ {∞}
is defined by vp (0) = ∞ and
vp (a) = ordp (a)
with
a = pordp (a)
b
c
and gcd(bc, p) = 1.
Definition 2.2.3. The subgroup v(K ∗ ) ⊆ R is called the value group of v.
Proposition 2.2.4. Let |·| be a nonarchimedean valuation of field K. Then v(x) = − log|x|
for x 6= 0 and v(0) = ∞ defines an exponential valuation v of K.
Proof. Later.
Corollary 2.2.5. Let v be an exponential valuation of field K. Then
|x| = q −v(x)
for some fixed real number q > 1 defines a nonarchimedean valuation |·| of K.
Proof. Later.
Definition 2.2.6. Two exponential valuations v1 , v2 of K are called equivalent if v1 = sv2
for some real number s > 0.
Remark 2.2.7. The definition of equivalent exponential valuations is consistent with the
notion of equivalent absolute values
|x|1 = |x|s2
for every x ∈ K.
2.3. Valuation Rings.
Proposition 2.3.1. Let v be an exponential valuation of field K and |·| the corresponding
nonarchimedean valuation of K defined in Corollary ??. Then
O
= {x ∈ K | v(x) ≥ 0} = {x ∈ K | |x| ≤ 1}
is a local ring with group of units
O
∗
= {x ∈ K | v(x) = 0} = {x ∈ K | |x| = 1}
and unique maximal ideal
p = {x ∈ K | v(x) > 0} = {x ∈ K | |x| < 1}.
Proof. Later.
7
Example 2.3.2. Let Qp be the completion of Q with respect to the p-adic absolute value |·|p
for some rational prime p. Qp is called the field of p-adic numbers and is the field of fractions
of the ring of p-adic integers
Zp = {x ∈ Qp | |x|p ≤ 1}.
Zp is also the completion of Z with respect to |·|p . The group of units of Zp is
Z∗p = {x ∈ Qp | |x|p = 1}
and the unique maximal ideal
pZp = {x ∈ Qp | |x|p < 1}.
Corollary 2.3.3. The local ring O in Proposition ?? is a integrally closed integral domain
with field of fractions K. For every x ∈ K, either x ∈ O or x−1 ∈ O. So the unique maximal
ideal in Proposition ?? is given by
p = {x ∈ O | x−1 ∈
/ O}.
Proof. Later.
Definition 2.3.4. A ring with properties as described by Corollary ?? is called a valuation
ring O and the field O/p the residue class field of O.
Definition 2.3.5. An exponential valuation v of K is called discrete if the value group
v(K ∗ ) ⊆ Z and normalized if v(K ∗ ) = Z.
Definition 2.3.6. Let v be a normalized exponential valuation of K and O the associated
valuation ring. An element π ∈ O such that v(π) = 1 is called a prime element of O.
Proposition 2.3.7. If v is a normalized exponential valuation of K and
valuation ring, then every element x ∈ K ∗ has a unique representation
O
the associated
x = uπ m
with m ∈ Z, u ∈ O∗ , and π ∈ O a prime element.
Proof. Later.
Proposition 2.3.8. If v is a discrete exponential valuation of K, then
O
= {x ∈ K | v(x) ≥ 0}
is a principal ideal domain and so a discrete valuation ring. Also, if v is normalized and
π ∈ O is a prime element, then the ideals of O are given by
pn = π n O = {x ∈ K | v(x) ≥ n},
n≥0
and
pn /pn+1 ∼
= O/p.
Proof. Later.
8
Example 2.3.9. The ideals of Zp are the principal ideals
pn Zp = {x ∈ Qp | vp (x) ≥ n},
n≥0
and
Zp /pn Zp ∼
= Z/pn Z.
2.4. Extensions of Dedekind Domains.
Notation 2.4.1. By Proposition ??, every prime ideal p of
way into a product of prime ideals of O,
O
decomposes in O in a unique
pO = Pe11 · · · Perr .
Definition 2.4.2. A prime ideal P of O in the decomposition of pO for a prime ideal p in
O is said to lie over p since p = P ∩ O. This is also written as P|p and P is called a prime
divisor of p.
Definition 2.4.3. The exponent ei in Notation ?? is called the ramification index and the
degree of the residue class field extension fi = [O/Pi : O/p] is called the inertia degree of Pi
over p.
Proposition 2.4.4. If L|K is a separable finite extension of K of degree n, then
r
X
ei fi = n.
i=1
Proof. Later.
Definition 2.4.5. Using Notation ??, the prime ideal p is said to split completely in L if
r = n = [L : K], in which case ei = fi = 1 for each i. The prime ideal p is called nonsplit if
r = 1 so that there is only one prime ideal P of L over p.
Definition 2.4.6. Using Notation ??, the prime ideal Pi is called unramified over O if ei = 1
and the residue class field extension O/Pi |O/p is separable. Else, Pi is called ramified. If
the prime ideal Pi is ramified and fi = 1, then it is called totally ramified. Also, the prime
ideal p is called unramified if all Pi are unramified. Else, p is called ramified. The extension
L|K is called unramified if all prime ideals p of O are unramified in L.
Proposition 2.4.7. If L|K is a separable finite extension of K, then there are only finitely
many prime ideals of O which are ramified in L.
Proof. Later.
2.5. Completions.
9
Definition 2.5.1. A valued field (K, |·|) is called complete if every Cauchy sequence (an ) in
K converges with respect to |·| to an element a ∈ K, i.e.,
lim |an − a| = 0.
n→∞
Notation 2.5.2. Let (K, |·|) be a valued field. Then the completion of the valued field is
b |·|) and is a complete valued field.
denoted (K,
Example 2.5.3. The valued field (R, |·|∞ ) is the completion of (Q, |·|∞ ) and (C, |·|∞ ) the
completion of (R, |·|∞ ). Also, (Qp , |·|p ) is the completion of valued field (Q, |·|p ) for every
rational prime p.
Theorem 2.5.4. (Ostrowski’s Theorem) If (K, |·|) is a complete valued field with respect to
an archimedean valuation, then there is an isomorphism
σ:K→F
with F = R or C such that
|a| = |σ(a)|s
for all a ∈ K
for some fixed s ∈ (0, 1].
Proof. Later.
Proposition 2.5.5. Let (K, v) be a valued field with valuation ring O and unique maximal
b v̂) is the completion of (K, v) with valuation ring O
b and unique maximal ideal
ideal p. If (K,
b
p, then the residue class fields
b/b
O
p∼
= O/p.
If v is discrete, then
b/b
O
pn ∼
= O/pn , n ≥ 1.
Proof. Later.
Theorem 2.5.6. If (K, |·|1 ) is a complete valued field, then the valuation |·|1 may be
extended in a unique way to a valuation |·|2 of any algebraic extension L|K. This valuation
is given by
q
|α|2 =
n
|NL|K (α)|1
with n = [L : K] and makes (L, |·|2 ) into a complete valued field.
Proof. Later.
Corollary 2.5.7. If v is the exponential valuation of K associated with the valuation |·|1 in
Theorem ?? and defined as in Proposition ??, then v extends uniquely to the exponential
valuation w of L given by
1
w(α) = v(NL|K (α))
n
with n = [L : K].
10
Proof. By Proposition ?? and Theorem ??,
q
1
w(α) = − log|α|2 = − log n |NL|K (α)|1 = − log|NL|K (α)|1 .
n
But
v(NL|K (α)) = − log|NL|K (α)|1
and so
1
1
w(α) = − log|NL|K (α)|1 = v(NL|K (α)).
n
n
2.6. Local Fields.
Definition 2.6.1. A finite field extension of Q is called a global field.
Definition 2.6.2. A complete valued field with a discrete valuation and finite residue class
field is called a local field.
Example 2.6.3. A completion of a global field is a local field.
Notation 2.6.4. The normalized exponential valuation of a local field is denoted by vp and
the normalized absolute value is given by
|x|p = q −vp (x)
with q the size of the residue class field.
Proposition 2.6.5. The local fields are the finite extensions of the complete valued fields
Qp .
Proof. Later.
Definition 2.6.6. The finite extensions of Qp are called p-adic number fields.
2.7. Extensions of Valuations.
Notation 2.7.1. Let K be a field. The exponential valuation of K is denoted by v. The
corresponding absolute value is denoted by |·|v and the completion Kv . Let K v be an
algebraic closure of Kv . The canonical extension of v to Kv is also denoted by v and the
unique extension of v to K v by v.
Proposition 2.7.2. Let L|K be an algebraic extension of degree n. Select one of the n
distinct K-embeddings
σ : L → Kv.
Since the valuation
v : K v → R ∪ {∞},
then
w = v ◦ σ : L → R ∪ {∞}
11
is an extension of valuation v to L. In terms of absolute values,
|x|w = |σ(x)|v .
Proof. Later.
Corollary 2.7.3. The K-embedding
σ : L → Kv
in Proposition ?? is continuous as a map between topological spaces defined by
|x|w = |σ(x)|v
and extends in a unique way to a continuous K-embedding
σ : Lw → K v .
Proof. Later.
Remark 2.7.4. The field Lw in Corollary ?? is the completion of (L, |·|w ) when L|K is a finite
extension.
Proposition 2.7.5. The canonical extension of the valuation w to Lw is the unique extension
of the valuation v from Kv to the extension Lw |Kv . So
Lw = LKv .
Proof. Later.
Corollary 2.7.6. If the field extension Lw |Kv in Proposition ?? has degree n, then the
relation between the absolute values of the complete valued fields (Kv , |·|v ) and (Lw , |·|w ) is
q
|x|w = n |NLw |Kv (x)|v .
Proof. Later.
Definition 2.7.7. The method of passing from the global extension L|K to the local extension
Lw |Kv as detailed in these Propositions/Corollaries is called the local-to-global principle.
Proposition 2.7.8. Let ρ : K v → K v be an automorphism of K v that fixes Kv . Then
σ0 = ρ ◦ σ : L → K v
is a new K-embedding of L called the conjugate to σ over Kv .
Proof. Later.
Theorem 2.7.9. (Extension Theorem) Let L|K be an algebraic field extension and v a
valuation of K. Then
(i) Every extension w of the valuation v is the composite v ◦ σ for some K-embedding
σ : L → Kv.
12
(ii) Two extensions v ◦ σ and v ◦ σ 0 are equal if and only if σ and σ 0 are conjugate over
Kv .
Proof. Later.
Example 2.7.10. Let K|Q be an algebraic field extension of degree n and v = vp the p-adic
valuation of Q. Then Qv = Qp and Qv = Qp = Cp . So every extension w of the valuation
vp to K is the p-adic valuation w = vp for some prime ideal p of OK such that p|pOK . By
Theorem ??, there are at most n prime ideals dividing pOK and so at most n extensions
vp |vp .
Remark 2.7.11. Example ?? is a restatement of Proposition ?? and Theorem ?? is a generalization
of the Proposition.
Notation 2.7.12. Let L|K be a finite algebraic extension. If v is a valuation of K and w is
an extension of v to L, then denote this relation by w|v.
Proposition 2.7.13. Let L|K be a separable finite extension of degree n. Then
X
n = [L : K] =
[Lw : Kv ]
w|v
and
T rL|K (x) =
X
T rLw Kv (x),
NL|K (x) =
Y
w|v
NLw |Kv (x).
w|v
Proof. Later.
Definition 2.7.14. Using Notation ??, the valuation ring Ov and unique maximal ideal pv
form the residue class field Ov /pv of Ov . Likewise, Ow /pw is the residue class field of Ow .
Since v(K ∗ ) ⊆ w(L∗ ) for the value groups, then the index
ew = (w(L∗ ) : v(K ∗ )),
called the ramification index of w|v, is well defined. Also,
Ov /pv
⊆ Ow /pw and so the degree
fw = [Ow /pw : Ov /pv ],
called the inertia degree of w|v, is well defined.
Remark 2.7.15. For the inclusion of the residue class fields,
Ov /pv
∼
= f ⊆ Ow /pw .
Proposition 2.7.16. If L|K is a separable finite extension with a discrete valuation v of K,
then
X
ew fw = [L : K].
w|v
Proof. Later.
13
Remark 2.7.17. From Propositions ?? and ??,
X
X
ew fw = [L : K] =
[Lw : Kv ] =⇒ [Lw : Kv ] = ew fw
w|v
w|v
so the degree of the field extension Lw |Kv is ew fw .
Definition 2.7.18. The equation in Proposition ?? is called the fundamental identity of
valuation theory.
Example 2.7.19. If K is the field of fractions of a Dedekind domain
valuation vp of K for every prime ideal p of O is defined by
O,
then the p-adic
vp (a) = vp
with
Y
(a) =
pvp .
p
By Proposition ??, the valuation ring of vp is the localization
of O in L and
pO = Pe11 · · · Perr
Op .
If O is the integral closure
is the prime decomposition of p in L, then the valuation vp extends to L by
wi =
1
vP
ei i
for each i. The exponents ei are the ramification indices and fi = [O/Pi : O/p] the inertia
degrees of Pi over p. Once again, the fundamental identity
r
X
ei fi = [L : K]
i=1
holds with the degree of each field extension Lwi |Kvp given by
ei fi = [Lwi : Kvp ].
Definition 2.7.20. Let K be an algebraic number field with p a prime ideal of OK and
p = p ∩ Z a rational prime. The unique renormalized p-adic absolute value of K extending
the p-adic absolute value of Q
k·kp : K → R
is defined by k0kp = 0 and for x 6= 0,
kxkp = p
−
ordp (x)
ep/p
with
b
(x) = xOK = pordp (x) ,
c
p - bc,
and ep/p = ordp (p).
Proposition 2.7.21. If K is an algebraic number field with p a prime ideal of OK , then
k·kp and |·|p are equivalent valuations of K.
14
Proof. Fix a prime ideal p of OK and let p = p ∩ Z be the unique rational prime. Since
N (p) = pfp/p
with fp/p = [OK /p : Z/(p)],
∗
then for any x ∈ K ,
|x|p = N (p)− ordp (x) = (pfp/p )− ordp (x) = p−fp/p ordp (x) .
So
kxkp = p
−
ordp (x)
ep/p
e
=⇒ |x|p = kxkpp/p
fp/p
.
But
ep/p = ordp (p) > 0 =⇒ ep/p fp/p = [Kp : Qp ] > 0
and so by Proposition ??, k·kp and |·|p are equivalent valuations of K.
b a finite extension of Kp ,
Corollary 2.7.22. If (Kp , k·kp ) is a complete valued field and L
b
then k·kp extends uniquely to L.
Proof. Later.
b a finite extension of Kp ,
Proposition 2.7.23. If (Kp , k·kp ) is a complete valued field and L
b = Lq for some finite extension L of K with q|p.
then L
Proof. Later.
Remark 2.7.24. The normalized p-adic valuation holds true in the product formula but may
not be extended. The renormalized p-adic valuation, however, may be extended to any field
extension but the product formula fails. Both valuations are equivalent and the fundamental
identity of valuation theory holds.
2.8. More Examples.
√
Example 2.8.1. Let K = Q(i 5). The ramified primes are 2 and 5. Specifically,
√
h2i = h2, 1 + i 5i2
and
√
h5i = hi 5i2 .
√
So by Definition ??, for p = h2, 1 + i 5i,
√
1
1
k2, 1 + i 5kp = 2− 2 = √ ,
2
2
1
k2kp = 2− 2 =
2
√
and for p = hi 5i,
√
1
1
ki 5kp = 5− 2 = √ ,
5
2
1
k5kp = 5− 2 = .
5
15
Since
p−1
−1
5
5
−5
=
= (−1) 2
,
p
p
p
p
then
−5
= −1 for p ≡ 11, 13, 17, 19 (mod 20)
p
and so these primes remain prime in K. So for each of these prime ideals hpi,
1
kpkhpi = p−1 = .
p
Finally, for
−5
= 1 for p ≡ 1, 3, 7, 9 (mod 20)
p
so each of these principal ideals hpi factor into a product of two unique prime ideals of K.
Specifically,
and
√
So for p = h3, 1 + i 5i,
√
and for p = h3, 1 − i 5i,
√
Also, for p = h7, 3 + i 5i,
√
and for p = h7, 3 − i 5i,
√
√
h3i = h3, 1 + i 5ih3, 1 − i 5i
√
√
h7i = h7, 3 + i 5ih7, 3 − i 5i.
√
1
k3, 1 + i 5kp = 3−1 = ,
3
√
0
k3, 1 − i 5kp = 3 = 1,
1
k3kp = 3−1 = ,
3
√
1
k3, 1 − i 5kp = 3−1 = ,
3
√
0
k3, 1 + i 5kp = 3 = 1,
1
k3kp = 3−1 = .
3
√
1
k7, 3 + i 5kp = 7−1 = ,
7
√
k7, 3 − i 5kp = 70 = 1,
1
k7kp = 7−1 = ,
7
√
1
k7, 3 − i 5kp = 7−1 = ,
7
√
0
k7, 3 + i 5kp = 7 = 1,
1
k7kp = 7−1 = .
7
16