’’ ’’ ’’ ’’

¦ ¦ ¦

¦ ¦µ ¦

2 Sn

1 ’ ’ ’ F —2 ’ ’ ’ F— F — /F —2

’’ ’’ ’’’

’’ ’ ’ ’ 1.

’’

We also de¬ne the group of spinor norms:

Sn(A, σ, f ) = { µ(γ) | γ ∈ “(A, σ, f ) } ‚ F — ,

so Sn O+ (A, σ, f ) = Sn(A, σ, f )/F —2 , and the spin group:

Spin(A, σ, f ) = { γ ∈ “(A, σ, f ) | µ(γ) = 1 } ‚ “(A, σ, f ).

In the split case, if (A, σ, f ) = EndF (V ), σq , fq for some nonsingular quadratic

space (V, q) of even dimension, the standard identi¬cations associated to q yield

Spin(A, σ, f ) = Spin(V, q) = { c ∈ “+ (V, q) | „ (c) · c = 1 },

where „ is the canonical involution of C(V, q) which is the identity on V . From

the description of the spinor norm in Scharlau [?, Chap. 9, §3], it follows that the

group of spinor norms Sn(V, q) = Sn(A, σ, f ) consists of the products of any even

number of represented values of q.

§13. QUADRATIC PAIRS 187

The vector representation χ induces by restriction a homomorphism

Spin(A, σ, f ) ’ O+ (A, σ, f )

which we also denote χ.

(13.31) Proposition. The vector representation χ ¬ts into an exact sequence:

χ Sn

1 ’ {±1} ’’ Spin(A, σ, f ) ’’ O+ (A, σ, f ) ’’ F — /F —2 .

Proof : This follows from the exactness of the top sequence in (??) and the de¬nition

of Sn.

Assume now deg A = 2m ≥ 4. We may then use the extended Cli¬ord group

„¦(A, σ, f ) to de¬ne an analogue of the spinor norm on the group PGO+ (A, σ, f ), as

we proceed to show. The map S de¬ned below may be obtained as a connecting map

in a cohomology sequence, see §??. Its target group is the ¬rst cohomology group of

the absolute Galois group of F with coe¬cients in the center of the algebraic group

Spin(A, σ, f ). The approach we follow in this subsection does not use cohomology,

but since the structure of the center of Spin(A, σ, f ) depends on the parity of m,

we divide the construction into two parts, starting with the case where the degree

of A is divisible by 4. As above, we let Z = Z(A, σ, f ) be the center of the Cli¬ord

algebra C(A, σ, f ).

(13.32) De¬nition. Assume deg A ≡ 0 mod 4. We de¬ne a homomorphism

S : PGO+ (A, σ, f ) ’ Z — /Z —2

as follows: for g · F — ∈ PGO+ (A, σ, f ), pick ω ∈ „¦(A, σ, f ) such that χ (ω) = g · F —

and set

S(g · F — ) = σ(ω)ω · Z —2 = µ(ω) · Z —2 .

Since ω is determined by g · F — up to a factor in Z — and σ is of the ¬rst kind, by

(??), the element µ(ω)ω · Z —2 depends only on g · F — . The map S thus makes the

following diagram commute:

χ

1 ’ ’ ’ Z — ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦ ¦

¦ ¦µ ¦

2 S

1 ’ ’ ’ Z —2 ’ ’ ’ Z— Z — /Z —2

’’ ’’ ’’’

’’ ’ ’ ’ 1.

’’

Besides the formal analogy between the de¬nition of S and that of the spinor

norm Sn, there is also an explicit relationship demonstrated in the following propo-

sition:

(13.33) Proposition. Assume deg A ≡ 0 mod 4. Let

π : O+ (A, σ, f ) ’ PGO+ (A, σ, f )

be the canonical map. Then, the following diagram is commutative with exact rows:

µ

π

O+ (A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ F — /F —2

’’ ’’

¦ ¦

¦ ¦

Sn S

NZ/F

F — /F —2 Z — /Z —2 ’ ’ ’ F — /F —2 .

’’’

’’ ’’

188 III. SIMILITUDES

Proof : Consider g ∈ O+ (A, σ, f ) and γ ∈ “(A, σ, f ) such that χ(γ) = g. We then

have Sn(g) = µ(γ) · F —2 . On the other hand, we also have χ (γ) = g · F — , hence

S(g · F — ) = µ(γ) · Z —2 . This proves that the left square is commutative.

Consider next g · F — ∈ PGO+ (A, σ, f ) and ω ∈ „¦(A, σ, f ) such that χ (ω) =

g · F — , so that S(g · F — ) = µ(ω) · Z —2 . By (??) we have µ(ω) · F — = κ(ω) and, by

(??), NZ/F κ(ω) = µ(g) · F —2 . Therefore, NZ/F S(g · F — ) = µ(g) · F —2 and the

right square is commutative.

Exactness of the lower sequence is a consequence of Hilbert™s Theorem 90 (??):

if z ∈ Z — is such that zι(z) = x2 for some x ∈ F — , then NZ/F (zx’1 ) = 1, hence

by (??) there exists some y ∈ Z — such that zx’1 = ι(y)y ’1 . (Explicitly, we may

take y = t + xz ’1 ι(t), where t ∈ Z is such that t + xz ’1 ι(t) is invertible.) Then

zy 2 = xNZ/F (y), hence z · Z —2 lies in the image of F — /F —2 .

Note that the spin group may also be de¬ned as a subgroup of the extended

Cli¬ord group: for deg A ≡ 0 mod 4,

Spin(A, σ, f ) = { ω ∈ „¦(A, σ, f ) | µ(ω) = 1 }.

Indeed, (??) shows that the right side is contained in “(A, σ, f ).

The restriction of the homomorphism

χ : „¦(A, σ, f ) ’ PGO+ (A, σ, f )

to Spin(A, σ, f ), also denoted χ , ¬ts into an exact sequence:

(13.34) Proposition. Assume deg A ≡ 0 mod 4. The sequence

χ S

1 ’ µ2 (Z) ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ Z — /Z —2

’ ’

is exact, where µ2 (Z) = { z ∈ Z | z 2 = 1 }.

Proof : Let g · F — be in the kernel of S. Then there exists ω ∈ „¦(A, σ, f ), with

σ(ω)ω = z 2 for some z ∈ Z — , such that χ (ω) = g · F — . By replacing ω by ωz ’1 ,

we get ω ∈ Spin(A, σ, f ). Exactness at Spin(A, σ, f ) follows from the fact that

Z — © Spin(A, σ, f ) = µ2 (Z) in „¦(A, σ, f ).

In the case where deg A ≡ 2 mod 4, the involution σ is of the second kind,

hence µ(ω) ∈ F — for all ω ∈ „¦(A, σ, f ). The rˆle played by µ in the case where

o

deg A ≡ 0 mod 4 is now played by a map which combines µ and κ.

Consider the following subgroup U of F — — Z — :

U = { (±, z) | ±4 = NZ/F (z) } ‚ F — — Z —

and its subgroup U0 = { NZ/F (z), z 4 | z ∈ Z — }. Let22

H 1 (F, µ4 [Z] ) = U/U0 ,

and let [±, z] be the image of (±, z) ∈ U in H 1 (F, µ4 [Z] ).

For ω ∈ „¦(A, σ, f ), let k ∈ Z — be a representative of κ(ω) ∈ Z — /F — . The

element kι(k)’1 is independent of the choice of the representative k and we de¬ne

µ— (ω) = µ(ω), kι(k)’1 µ(ω)2 ∈ U.

22 It

will be seen in Chapter ?? (see (??)) that this factor group may indeed be regarded as

a Galois cohomology group if char F = 2. This viewpoint is not needed here, however, and this

de¬nition should be viewed purely as a convenient notation.

§13. QUADRATIC PAIRS 189

For z ∈ Z — , we have κ(z) = z 2 · F — and µ(z) = NZ/F (z), hence

µ— (z) = NZ/F (z), z 4 ∈ U0 .

(13.35) De¬nition. Assume deg A ≡ 2 mod 4. De¬ne a homomorphism

S : PGO+ (A, σ, f ) ’ H 1 (F, µ4 [Z] )

as follows: for g · F — ∈ PGO+ (A, σ, f ), pick ω ∈ „¦(A, σ, f ) such that χ (ω) = g · F —

and let S(g · F — ) be the image of µ— (ω) in H 1 (F, µ4 [Z] ). Since ω is determined up

to a factor in Z — and µ— (Z — ) ‚ U0 , the de¬nition of S(g · F — ) does not depend on

the choice of ω. In other words, S is the map which makes the following diagram

commute:

χ

1 ’ ’ ’ Z — ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦ ¦

¦ µ— ¦ µ— ¦

S

’ ’ ’ H 1 (F, µ4 [Z] ) ’ ’ ’ 1.

1 ’ ’ ’ U0 ’ ’ ’

’’ ’’ U ’’ ’’

In order to relate the map S to the spinor norm, we de¬ne maps i and j which

¬t into an exact sequence

j

i

F — /F —2 ’ H 1 (F, µ4 [Z] ) ’ F — /F —2 .

’ ’

For ± · F —2 ∈ F — /F —2 , we let i(± · F —2 ) = [±, ±2 ]. For [±, z] ∈ H 1 (F, µ4 [Z] ),

we pick z0 ∈ Z — such that ±’2 z = z0 ι(z0 )’1 , and let j[±, z] = NZ/F (z0 ) · F —2 ∈

F — /F —2 . If NZ/F (z0 ) = β 2 for some β ∈ F — , then we may ¬nd z1 ∈ Z — such that

z0 β ’1 = z1 ι(z1 )’1 . It follows that ±’2 z = z1 ι(z1 )’2 , hence

2

(±, z) = ±NZ/F (z1 )’1 , ±2 NZ/F (z1 )’2 · NZ/F (z1 ), z1 ,

4

and therefore [±, z] = i ±NZ/F (z1 )’1 · F —2 . This proves exactness of the sequence

above.

(13.36) Proposition. Assume deg A ≡ 2 mod 4. Let

π : O+ (A, σ, f ) ’ PGO+ (A, σ, f )

be the canonical map. Then, the following diagram is commutative with exact rows:

µ

π

O+ (A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ F — /F —2

’’ ’’

¦ ¦

¦ ¦

Sn S

j

i

F — /F —2 ’ ’ ’ H 1 (F, µ4 [Z] ) ’ ’ ’ F — /F —2 .

’’ ’’

Proof : Let g ∈ O+ (A, σ, f ) and let γ ∈ “(A, σ, f ) be such that χ(γ) = g. We then

have Sn(g) = µ(γ) · F —2 . On the other hand, we also have κ(γ) = 1, by (??), hence

µ— (γ) = µ(γ), µ(γ)2 = i µ(γ) · F —2 .

Since χ (γ) = g · F — , this proves commutativity of the left square.

Consider next g · F — ∈ PGO+ (A, σ, f ) and ω ∈ „¦(A, σ, f ) such that χ (ω) =

g · F — . We have j —¦ S(g · F — ) = NZ/F (k) · F — , where k ∈ Z — is a representative of

κ(ω) ∈ Z — /F — . Proposition (??) shows that NZ/F (k) · F —2 = µ(g · F — ), hence the

right square is commutative. Exactness of the bottom row was proved above.

190 III. SIMILITUDES

As in the preceding case, the spin group may also be de¬ned as a subgroup of

„¦(A, σ, f ): we have for deg A ≡ 2 mod 4,

Spin(A, σ, f ) = { ω ∈ „¦(A, σ, f ) | µ— (ω) = (1, 1) },

since (??) shows that the right-hand group lies in “(A, σ, f ). Furthermore we have

a sequence corresponding to the sequence (??):

(13.37) Proposition. Assume deg A ≡ 2 mod 4 and deg A ≥ 4. The sequence

χ S

1 ’ µ4 [Z] (F ) ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ H 1 (F, µ4 [Z] ),

’ ’

is exact, where µ4 [Z] (F ) = { z ∈ Z — | z 4 = 1 and ι(z)z = 1 }.

Proof : As in the proof of (??) the kernel of S is the image of Spin(A, σ, f ) under

χ . Furthermore we have by (??)

Z — © Spin(A, σ, f ) = { z ∈ Z — | z 2 ∈ F — and σ(z)z = 1 } = µ4 [Z] (F )

in „¦(A, σ, f ).

13.C. Multipliers of similitudes. This section is devoted to a generalization

of Dieudonn´™s theorem on the multipliers of similitudes (??). As in the preceding

e

sections, let (σ, f ) be a quadratic pair on a central simple algebra A of even degree

over an arbitrary ¬eld F , and let Z = Z(A, σ, f ) be the center of the Cli¬ord algebra

C(A, σ, f ). The nontrivial automorphism of Z/F is denoted by ι.

For ± ∈ F — , let (Z, ±)F be the quaternion algebra Z • Zj where multiplication

is de¬ned by jz = ι(z)j for z ∈ Z and j 2 = ±. In other words,

F [X]/(X 2 ’ δ);

(δ, ±)F if char F = 2 and Z

(Z, ±)F =

F [X]/(X 2 + X + δ).

[δ, ±)F if char F = 2 and Z

(Compare with §??).

(13.38) Theorem. Let g ∈ GO(A, σ, f ) be a similitude of (A, σ, f ).

(1) If g is proper, then Z, µ(g) F splits.

(2) If g is improper, then Z, µ(g) F is Brauer-equivalent to A.

When A splits, the algebra Z, µ(g) F splits in each case, so µ(g) is a norm

from Z/F for every similitude g. We thus recover Dieudonn´™s theorem (??).

e

In the case where g is proper, the theorem follows from (??) (or, equivalently,

from (??) and (??)). For the rest of this section, we ¬x some improper similitude g.

According to (??), the automorphism C(g) of C(A, σ) then restricts to ι on Z,

∼

so C(g) induces a Z-algebra isomorphism C(A, σ, f ) ’ ι C(A, σ, f ) by mapping

’

c ∈ C(A, σ, f ) to ι C(g)(c) ∈ ι C(A, σ, f ). When we view C(A, σ, f ) as a left

Z-module, we have the canonical isomorphism

C(A, σ, f )op —Z C(A, σ, f ) = EndZ C(A, σ, f )

which identi¬es cop — c2 with the endomorphism de¬ned by

1

op

c c1 —c2

= c1 cc2 for c, c1 , c2 ∈ C(A, σ, f ).

We then have Z-algebra isomorphisms:

EndZ C(A, σ, f ) = C(A, σ, f )op —Z C(A, σ, f ) ι

C(A, σ, f )op —Z C(A, σ, f ).

§13. QUADRATIC PAIRS 191

The embedding ν of ι C(A, σ, f )op —Z C(A, σ, f ) into the endomorphism algebra of

the Cli¬ord bimodule B(A, σ, f ) (see (??)) yields an embedding

νg : EndZ C(A, σ, f ) ’ EndA B(A, σ, f )

de¬ned by

op

xνg (c1 —c2 )

= C(g)(c1 ) — x · c2 for c1 , c2 ∈ C(A, σ, f ), x ∈ B(A, σ, f ).

Let γg ∈ EndF C(A, σ, f ) be the endomorphism C(g ’1 ), i.e.,

cγg = C(g ’1 )(c) for c ∈ C(A, σ, f ).

Since g is improper, γg is not Z-linear, but ι-semilinear. Thus Int(γg ), which maps

’1

f ∈ EndZ C(A, σ, f ) to γg —¦ f —¦ γg , is an automorphism of EndZ C(A, σ, f ).

De¬ne an F -algebra Eg as follows:

Eg = EndZ C(A, σ, f ) • EndZ C(A, σ, f ) · y

where y is subject to the following relations:

’1

yf = (γg —¦ f —¦ γg )y for f ∈ EndZ C(A, σ, f ),

y 2 = µ(g)γg .

2

The algebra Eg is thus a generalized cyclic algebra (see Albert [?, Theorem 11.11],

Jacobson [?, § 1.4]); the same arguments as for the usual cyclic algebras show E g

is central simple over F .

(13.39) Proposition. The homomorphism νg extends to an isomorphism of F -

algebras:

∼

νg : Eg ’ EndA B(A, σ, f )

’

by mapping y to the endomorphism β(g) of (??).

Proof : Since Eg and EndA B(A, σ, f ) are central simple F -algebras of the same

dimension, it su¬ces to show that the map de¬ned above is a homomorphism, i.e.,

that β(g) satis¬es the same relations as y:

’1

β(g) —¦ νg (f ) = νg (γg —¦ f —¦ γg ) —¦ β(g) for f ∈ EndZ C(A, σ, f ),

(13.40)

β(g)2 = µ(g)νg (γg ).

2

It su¬ces to check these relations over an extension of the base ¬eld. We may

thus assume that F is algebraically closed and (A, σ, f ) = EndF (V ), σq , fq for

some nonsingular quadratic space (V, q). Choose » ∈ F — satisfying »2 = µ(g).

Then q »’1 g(v) = q(v) for all v ∈ V , hence there is an automorphism of C(V, q)

which maps v to »’1 g(v) for all v ∈ V . By the Skolem-Noether theorem we may

thus ¬nd b ∈ C(V, q)— such that

b · v · b’1 = »’1 g(v) for v ∈ V .

Then C(g) is the restriction of Int(b) to C0 (V, q) = C(A, σ, f ), hence γg = Int(b’1 )

and

γg = (b2 )op — b’2 ∈ C(A, σ, f )op —Z C(A, σ, f ).

2

On the other hand, for v, w1 , . . . , w2r’1 ∈ V ,

β(g)

= µ(g)r v — g ’1 (w1 ) · · · g ’1 (w2r’1 )

v — (w1 · · · w2r’1 )

= » v — b’1 · (w1 · · · w2r’1 ) · b .

192 III. SIMILITUDES

The equations (??) then follow by explicit computation.

To complete the proof of (??) we have to show that the algebra Eg is Brauer-

equivalent to the quaternion algebra Z, µ(g) F . As pointed out by A. Wadsworth,

this is a consequence of the following proposition:

(13.41) Proposition. Let S be a central simple F -algebra, let Z be a quadratic

Galois ¬eld extension of F contained in S, with nontrivial automorphism ι. Let

s ∈ S be such that Int(s)|Z = ι. Let E = CS (Z) and ¬x t ∈ F — . Let T be the

F -algebra with presentation

where yey ’1 = ses’1 for all e ∈ E, and y 2 = ts2 .

T = E • Ey,

Then M2 (T ) (Z, t) —F S.

Proof : Let j be the standard generator of (Z, t) with jzj ’1 = ι(z) for all z ∈ Z

and j 2 = t. Let R = (Z, t) —F S, and let

T = (1 — E) + (1 — E)y ‚ R, where y = j — s.

Then T is isomorphic to T , since y satis¬es the same relations as y. (That is, for

’1 2

= 1 — (ses’1 ) and y = 1 — ts2 .) By the double centralizer

any e ∈ E, y (1 — e)y

theorem (see (??)) R T —F Q where Q = CR (T ), and Q is a quaternion algebra

over F by a dimension count. It su¬ces to show that Q is split. For, then

R T —F Q T —F M2 (F ) M2 (T ).

Consider Z —F Z ‚ R; Z —F Z centralizes 1 — E and Int(y ) restricts to ι — ι on

Z—F Z. Now Z—F Z has two primitive idempotents e1 and e2 , since Z—F Z Z•Z.

The automorphisms Id — ι and ι — Id permute them, so ι — ι maps each ei to itself.

Hence e1 and e2 lie in Q since they centralize 1 — E and also y . Because e1 e2 = 0,

Q is not a division algebra, so Q is split, as desired.

(13.42) Remark. We have assumed in the above proposition that Z is a ¬eld.

The argument still works, with slight modi¬cation, if Z F — F .

As a consequence of Theorem (??), we may compare the group G(A, σ, f ) of

multipliers of similitudes with the subgroup G+ (A, σ, f ) of multipliers of proper

similitudes. Since the index of GO+ (A, σ, f ) in GO(A, σ, f ) is 1 or 2, it is clear

that either G(A, σ, f ) = G+ (A, σ, f ) or G+ (A, σ, f ) is a subgroup of index 2 in

G(A, σ, f ). If A is split, then

[GO(A, σ, f ) : GO+ (A, σ, f )] = [O(A, σ, f ) : O+ (A, σ, f )] = 2

and

G(A, σ, f ) = G+ (A, σ, f )

since hyperplane re¬‚ections are improper isometries (see (??)). If A is not split, we

deduce from (??):

(13.43) Corollary. Suppose that (σ, f ) is a quadratic pair on a central simple

F -algebra A of even degree. If A is not split, then O(A, σ, f ) = O+ (A, σ, f ) and

[GO(A, σ, f ) : GO+ (A, σ, f )] = [G(A, σ, f ) : G+ (A, σ, f )].

If G(A, σ, f ) = G+ (A, σ, f ), then A is split by Z = Z(A, σ, f ).

§14. UNITARY INVOLUTIONS 193

Proof : If g ∈ O(A, σ, f ) is an improper isometry, then (??) shows that A is Brauer-

equivalent to Z, µ(g) F , which is split since µ(g) = 1. This contradiction shows

that O(A, σ, f ) = O+ (A, σ, f ).

If [GO(A, σ, f ) : GO+ (A, σ, f )] = [G(A, σ, f ) : G+ (A, σ, f )], then necessarily

GO(A, σ, f ) = GO+ (A, σ, f ) and G(A, σ, f ) = G+ (A, σ, f ).

Therefore, A contains an improper similitude g, and µ(g) = µ(g ) for some proper

similitude g . It follows that g ’1 g is an improper isometry, contrary to the equality

O(A, σ, f ) = O+ (A, σ, f ).

Finally, if µ is the multiplier of an improper similitude, then (??) shows that

A is Brauer-equivalent to (Z, µ)F , hence it is split by Z.

(13.44) Corollary. If disc(σ, f ) is trivial, then G(A, σ, f ) = G+ (A, σ, f ).

Proof : It su¬ces to consider the case where A is not split. Then, if G(A, σ, f ) =

G+ (A, σ, f ), the preceding corollary shows that A is split by Z; this is impossible

if disc(σ, f ) is trivial, for then Z F — F .

§14. Unitary Involutions

In this section, we let (B, „ ) be a central simple algebra with involution of the

second kind over an arbitrary ¬eld F . Let K be the center of B and ι the nontrivial

automorphism of K/F .

We will investigate the group GU(B, „ ) of similitudes of (B, „ ) and the unitary

group U(B, „ ), which is the kernel of the multiplier map µ (see §??). The group

GU(B, „ ) has di¬erent properties depending on the parity of the degree of B. When

deg B is even, we relate this group to the group of similitudes of the discriminant

algebra D(B, „ ).

14.A. Odd degree.

(14.1) Proposition. If deg B is odd, the group G(B, „ ) of multipliers of simili-

tudes of (B, „ ) is the group of norms of K/F :

G(B, „ ) = NK/F (K — ).

Moreover, GU(B, „ ) = K — · U(B, „ ).

Proof : The inclusion NK/F (K — ) ‚ G(B, „ ) is clear, since K — ‚ GU(B, „ ) and

µ(±) = NK/F (±) for ± ∈ K — . In order to prove the reverse inclusion, let deg B =

2m + 1 and let g ∈ GU(B, „ ). By applying the reduced norm to the equation

„ (g)g = µ(g) we obtain

NK/F NrdB (g) = µ(g)2m+1 .

Therefore,

µ(g) = NK/F µ(g)’m NrdB (g) ∈ NK/F (K — ),

hence G(B, „ ) ‚ NK/F (K — ). This proves the ¬rst assertion.

The preceding equation shows moreover that µ(g)m NrdB (g)’1 g ∈ U(B, „ ).

Therefore, letting u = µ(g)m NrdB (g)’1 g and ± = µ(g)’m NrdB (g) ∈ K — , we get

g = ±u. Thus, GU(B, „ ) = K — · U(B, „ ).

Note that in the decomposition g = ±u above, the elements ± ∈ K — and

u ∈ U(B, „ ) are uniquely determined up to a factor in the group K 1 of norm 1

elements, since K — © U(B, „ ) = K 1 .

194 III. SIMILITUDES

14.B. Even degree. Suppose now that deg B = 2m and let g ∈ GU(B, „ ).

By applying the reduced norm to the equation „ (g)g = µ(g), we obtain

NK/F NrdB (g) = µ(g)2m ,

hence µ(g)m NrdB (g)’1 is in the group of elements of norm 1. By Hilbert™s The-

orem 90, there is an element ± ∈ K — , uniquely determined up to a factor in F — ,