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Realizing Algebraic Number Fields
1983In the paper [13], the authors studied the problem of realizing rational division algebras in a special way. Let D be a division algebra that is finite dimensional over the rational field Q. If p is a prime, we say that D is p-realizable when there is a p-local torsion free abelian group A whose rank is the dimension of D over Q, such that D is ...
R. S. Pierce, C. I. Vinsonhaler
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Boletín de la Sociedad Matemática Mexicana, 2015
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Gendron, T. M., Verjovsky, A.
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Gendron, T. M., Verjovsky, A.
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Number Fields and Number Rings
1977A number field is a subfield of ℂ having finite degree (dimension as a vector space) over ℚ. We know (see appendix 2) that every such field has the form ℚ[α] for some algebraic number α ∈ ℂ. If α is a root of an irreducible polynomial over ℚ, having degree n, then $$\mathbb{Q}[\alpha ] = \left\{ {{a_o} + {a_1}\alpha + \cdots + {a_{n - 1}}{\alpha ...
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Discriminants of Number Fields
Jahresbericht der Deutschen Mathematiker-VereinigungThis paper offers a nice overview on recent research on discriminants of algebraic number fields, especially on lower bounds for the root discriminant. Using (infinite) class field towers, one obtains number fields with small root discriminants. Such fields are needed for lattice-based cryptography, which is important for post-quantum cryptography. The
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2023
In "Non-Field Structure of the Reals, Projective System Preferred," it wasdemonstrated using standard variable algebra how the so called, "Real Num-bers," are actually a projective scheme, and do not truly form a, "field," asexceptions have to made for the multiplicative inverse when a variable equalszero, which is possible.
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In "Non-Field Structure of the Reals, Projective System Preferred," it wasdemonstrated using standard variable algebra how the so called, "Real Num-bers," are actually a projective scheme, and do not truly form a, "field," asexceptions have to made for the multiplicative inverse when a variable equalszero, which is possible.
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2012
This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number ...
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This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number ...
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1989
For the benefit of less experienced readers, we repeat some basic definitions of abstract algebra. A group is a pair (G, + G ) in which G is a set and + G is a closed associative binary operation on G for which the following hold: a) there exists an element 1 G of G, called the identity, which has the property that for any element a in G, we ...
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For the benefit of less experienced readers, we repeat some basic definitions of abstract algebra. A group is a pair (G, + G ) in which G is a set and + G is a closed associative binary operation on G for which the following hold: a) there exists an element 1 G of G, called the identity, which has the property that for any element a in G, we ...
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1993
An algebraic number field F is a finite extension field of the rational numbers ℚ. It can be generated by a root p of a monic irreducible polynomial $$f(t) = {{t}^{n}} + {{a}_{1}}{{t}^{{n - 1}}} + {\text{ }} \ldots + {{a}_{n}}\epsilon \mathbb{Z}[t]$$ , (27) where n is also called the degree of F.
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An algebraic number field F is a finite extension field of the rational numbers ℚ. It can be generated by a root p of a monic irreducible polynomial $$f(t) = {{t}^{n}} + {{a}_{1}}{{t}^{{n - 1}}} + {\text{ }} \ldots + {{a}_{n}}\epsilon \mathbb{Z}[t]$$ , (27) where n is also called the degree of F.
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