Witt kernels of bi-quadratic extensions in characteristic 2

Let κ be a field of characteristic 2. The author's previous results (Arch. Math. (1994)) are used to prove the excellence of quadratic extensions of κ. This in turn is used to determine the Witt kernel of a quadratic extension up to Witt equivalence. An example is given to show that Witt equivalence cannot be strengthened to isometry.

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On the Quadratic Extensions and the Extended Witt Ring of a Commutative Ring

Nagoya Mathematical Journal ◽

10.1017/s0027763000015348 ◽

1973 ◽

Vol 49 ◽

pp. 127-141 ◽

Cited By ~ 8

Author(s):

Teruo Kanzaki

Keyword(s):

Galois Group ◽

Commutative Ring ◽

Galois Extension ◽

Identity Element ◽

Quadratic Extension ◽

Crossed Product ◽

Witt Ring ◽

Division Rings ◽

Quadratic Extensions ◽

The Common

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

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Witt kernels of bi-quadratic extensions in characteristic 2: Addendum

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700034560 ◽

2004 ◽

Vol 70 (2) ◽

pp. 351-351

Author(s):

Hamza Ahmad

Keyword(s):

Quadratic Extensions ◽

Characteristic 2

In a recent referee's report reviewing [2], it was pointed out to me that Theorem 2.2 in [1] was already known in the literature, and is (originally) due to Mammone and Moresi in [3]. Also in [3], the authors establish the excellence of inseparable quadratic extensions in a shorter way than I presented in [1]. As the paper [3] was unknown to me at the time of submitting [1], I hope by this note to acknowledge and credit [3] for the result(s).

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W-Groups under Quadratic Extensions of Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-2000-036-0 ◽

2000 ◽

Vol 52 (4) ◽

pp. 833-848 ◽

Cited By ~ 1

Author(s):

Ján Mináč ◽

Tara L. Smith

Keyword(s):

General Setting ◽

Quadratic Extension ◽

Field Extension ◽

Galois Groups ◽

Real Field ◽

Quadratic Extensions ◽

Finite Subgroups ◽

Carry Over ◽

Rigid Element ◽

Pythagorean Field

AbstractTo each field F of characteristic not 2, one can associate a certain Galois group , the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paperwe investigate the connection between and (√a), where F(√a) is a proper quadratic extension of F. We obtain a precise description in the case when F is a pythagorean formally real field and a = −1, and show that the W-group of a proper field extension K/F is a subgroup of the W-group of F if and only if F is a formally real pythagorean field and K = F(√−1). This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when a is a double-rigid element in F. Some of these results carry over to the general setting.

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The L-Theory of Twisted Quadratic Extensions

Canadian Journal of Mathematics ◽

10.4153/cjm-1987-017-x ◽

1987 ◽

Vol 39 (2) ◽

pp. 345-364 ◽

Cited By ~ 18

Author(s):

Andrew Ranicki

Keyword(s):

Exact Sequence ◽

Normal Bundle ◽

Quadratic Extension ◽

Algebraic Version ◽

Quadratic Extensions ◽

Rings With Involution ◽

Index 2 ◽

Image Position

For surgery on codimension 1 submanifolds with non-trivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN∗(π′ → π), with π a group and π′ a subgroup of index 2, such that there is defined an exact sequence involving the ordinary L-groups of rings with involutionwith the superscript w signifying a different involution on Z[π]. Geometry was used in [13] to identifywith (α, u) an antistructure on Z[π′] in the sense of Wall [14]. The main result of this paper is a purely algebraic version of this identification, for any twisted quadratic extension of a ring with antistructure.

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Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II

Forum Mathematicum ◽

10.1515/forum-2017-0080 ◽

2018 ◽

Vol 30 (6) ◽

pp. 1487-1520 ◽

Cited By ~ 1

Author(s):

Sungmun Cho

Keyword(s):

Mass Formula ◽

Local Density ◽

Group Scheme ◽

Field Extension ◽

Integral Group ◽

Group Schemes ◽

Quadratic Extensions ◽

Local Densities ◽

Characteristic 2 ◽

Residue Characteristic

Abstract This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions {E/F} , where F is a finite unramified field extension of {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice {(L,H)} , when a base field is unramified over {\mathbb{Q}} at a prime {(2)} .

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Hecke’s Integral Formula for Relative Quadratic Extensions of Algebraic Number Fields

Nagoya Mathematical Journal ◽

10.1017/s0027763000009521 ◽

2008 ◽

Vol 189 ◽

pp. 139-154 ◽

Cited By ~ 2

Author(s):

Shuji Yamamoto

Keyword(s):

Zeta Function ◽

Eisenstein Series ◽

Algebraic Number ◽

Integral Formula ◽

Zeta Functions ◽

Quadratic Extension ◽

Number Fields ◽

Algebraic Number Fields ◽

Limit Formula ◽

Quadratic Extensions

AbstractLet K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F.

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On automorphism group of free quadratic extensions over a ring

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171284000107 ◽

1984 ◽

Vol 7 (1) ◽

pp. 103-108

Author(s):

George Szeto

Keyword(s):

Automorphism Group ◽

Galois Group ◽

Galois Extension ◽

Zero Divisor ◽

Quadratic Extension ◽

Normal Extension ◽

Quadratic Extensions ◽

The One

LetRbe a ring with1,ρan automorphism ofRof order2. Then a normal extension of the free quadratic extensionR[x,ρ]with a basis{1,x}overRwith anR-automorphism groupGis characterized in terms of the element(x−(x)α)forαinG. It is also shown by a different method from the one given by Nagahara that the order ofGof a Galois extensionR[x,ρ]overRwith Galois groupGis a unit inR. When2is not a zero divisor, more properties ofR[x,ρ]are derived.

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