A Necessary Condition for Gauss Period Normal Bases to Be the Same Normal Basis

2008 ◽  
Vol E91-A (4) ◽  
pp. 1229-1232
Author(s):  
Y. NOGAMI ◽  
R. NAMBA ◽  
Y. MORIKAWA
Keyword(s):  
Normal Basis ◽  
Necessary Condition ◽  
Gauss Period ◽  
Normal Bases

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

scholarly journals NORMAL BASES FOR MODULAR FUNCTION FIELDS

2017 ◽  
Vol 95 (3) ◽  
pp. 384-392
Author(s):  
JA KYUNG KOO ◽  
DONG HWA SHIN ◽  
DONG SUNG YOON
Keyword(s):  
Galois Extension ◽  
Meromorphic Functions ◽  
Function Fields ◽  
Modular Function ◽  
Normal Basis ◽  
Modular Curve ◽  
Normal Bases ◽  
Free Element

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let $\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve $X(N)$ of level $N$. We construct a completely free element in the extension $\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.

Consideration of Generating Suitable Parameters for Constructing type (h, m) Gauss Period Normal Basis

2021 ◽  
Author(s):  
Keiji Yoshimoto ◽  
Yuta Kodera ◽  
Takuya Kusaka ◽  
Yasuyuki Nogami
Keyword(s):  
Normal Basis ◽  
Gauss Period

A note on normal bases of ideals in sextic algebraic number field

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Viktor Dubovský ◽  
Juraj Kostra ◽  
Vladimir Lazar
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Basis ◽  
Normal Bases

AbstractLet K/Q be a cyclic tamely ramified extension of degree 6, then any ambiguous ideal of K has a normal basis.

scholarly journals Normal Bases in Galois Extensions of Number Fields

1969 ◽  
Vol 34 ◽  
pp. 153-167 ◽  
Author(s):  
S. Ullom
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Abstract Algebra ◽  
Normal Basis ◽  
Galois Module ◽  
Normal Bases ◽  
Ring Of Integers ◽  
Root Numbers ◽  
Fractional Ideal ◽  
Rings Of Integers

The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers OK of a number field was a free Z-module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F. A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.

On the Complexity of the Dual Bases of the Gaussian Normal Bases

2015 ◽  
Vol 22 (spec01) ◽  
pp. 909-922
Author(s):  
Alok Mishra ◽  
Rajendra Kumar Sharma ◽  
Wagish Shukla
Keyword(s):  
Finite Field ◽  
Normal Basis ◽  
Dual Basis ◽  
Normal Bases ◽  
Gaussian Normal Basis

In this paper, we study the complexity of the dual bases of the Gaussian normal bases of type (n, t), for all n and t = 3, 4, 5, 6, of 𝔽qn over 𝔽q and provide conditions under which the complexity of the Gaussian normal basis of type (n, t) is equal to the complexity of the dual basis over any finite field.

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