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Topic: Algebraic integers

	Integer - Wikipedia, the free encyclopedia
		The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.
		The integer q is called the quotient and r is called the remainder, resulting from division of a by b.
		Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers.
	en.wikipedia.org /wiki/Integer (800 words)

	Algebraic number - Wikipedia, the free encyclopedia
		If an algebraic number satisfies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number of degree n.
		In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
		The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.
	en.wikipedia.org /wiki/Algebraic_number (609 words)

	Algebraic Integers
		The degree of an algebraic number a is the minimal degree of a nonzero polynomial p(x) with integral coefficients having a as root.
		Algebraic integers are roots of a polynomial with integral coefficients and leading coefficient 1.
		Exercise An algebraic integer a of degree d is root of a polynomial with integral coefficients of degree d.
	www.win.tue.nl /~aeb/an/an.html (584 words)

	Algebraic integer - Wikipedia, the free encyclopedia
		In mathematics, an algebraic integer is an algebraic number that is a root of a monic polynomial (i.e.
		One may show that if P(x) is a non-monic primitive polynomial with integer coefficients that is irreducible over Q, then none of the roots of P are algebraic integers.
		In other words, the algebraic integers form a ring that is closed under the operation of extraction of roots.
	en.wikipedia.org /wiki/Algebraic_integer (200 words)

	Number theory - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well.
		In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
		Algebraic geometry, especially the theory of elliptic curves, may also be employed.
	www.sevenhills.us /project/wikipedia/index.php/Number_theory (1325 words)

	Algebraic integer - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		In mathematics, an algebraic integer is a complex number α that is a root of an equation
		All algebraic integers are therefore algebraic numbers, but it can be shown that not all algebraic numbers are algebraic integers.
		So all radical integers are algebraic integers but not all algebraic integers are radical integers.
	encyclopedia.worldsearch.com /algebraic_integer.htm (313 words)

	PlanetMath: algebraic number theory
		Algebraic number theory is the study of algebraic numbers, their properties and their applications.
		It is a well-known fact that the ring of integers of a number field is a Dedekind domain.
		This is version 29 of algebraic number theory, born on 2005-03-15, modified 2005-05-04.
	planetmath.org /encyclopedia/AlgebraicNumberTheory.html (936 words)

	PlanetMath: algebraic integer
		is called an algebraic integer of K it is the root of a monic polynomial with coefficients in
		This is version 6 of algebraic integer, born on 2001-10-15, modified 2004-07-29.
		You might want to add the simple generalization to an extension of the rationals that isn't all of C: Let K be an extension of Q. A number $\alpha \in K$ is called an algebraic integer of K it is the root of a monic polynomial with coefficients in $\mathbb{Z}$.
	planetmath.org /encyclopedia/AlgebraicInteger.html (136 words)

	Algebraic number (Site not responding. Last check: 2007-11-06)
		The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field.
		If K is a number field, its ring of integers is the subring of algebraic integers in K.
		Both the notions of algebraic number and algebraic integer may be usefully generalized to fields other than the complex numbers; see algebraic extension and integral closure.
	www.sciencedaily.com /encyclopedia/algebraic_number (445 words)

	Algebraic number (Site not responding. Last check: 2007-11-06)
		If an algebraic number satisifies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be analgebraic number of degree n.
		The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integersform a ring.
		Both the notions of algebraic number and algebraic integer may be usefully generalized to fields other than the complexnumbers; see algebraic extension and integral closure.
	www.therfcc.org /algebraic-number-19429.html (364 words)

	Algebraic Number Theory (Site not responding. Last check: 2007-11-06)
		An algebraic number of degree n is any root of a polynomial equation of degree n with integer coefficients that is not a root of any such equation of a lower degree.
		The ordinary rational number are algebraic integers of degree 1.
		Algebraic integers are also associated with Diophantine equations, Pell equations, and the studies of E E Kummer for that matter.
	www.risberg.ws /Hypertextbooks/Mathematics/Numbers/algebraic.htm (93 words)

	[No title]
		An algebraic integer is any number that is the root of some polynomial equation with integer coefficients whose leading coefficient is 1.
		Moreover, if we fix a finite algebraic extension F of the rationals, the set of all algebraic integers lying in F forms a subring of F called the "ring of integers" of F. Rings of algebraic integers have many properties analogous to those of the usual integers.
		This is the case for the ordinary integers: for each integer n, the set of all multiples of n is an ideal, and these are all the ideals.
	www-math.mit.edu /~tchow/mathstuff/CFT (1977 words)

	Number theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-06)
		This sense of the term arithmetic should not be confused with the branch of logic which studies arithmetic in the sense of formal systems.
		The virtue of the machinery employed -- Galois theory, field cohomology, class field theory, group representations and L-functions -- is that it allows to recover that order partly for this new class of numbers.
		The theory of numbers, a favorite study among the ancient Greeks, had its renaissance in the sixteenth and seventeenth centuries in the labors of Viète, Bachet de Meziriac, and especially Fermat.
	encyclopedia.learnthis.info /n/nu/number_theory.html (1115 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		A polynomial in one variable with algebraic integer coefficients can be factored into a product of linear terms, each eith algebraic integer coefficients (although you may need to go to an extension of the splitting field to do it).
		QED Now, we can improve on your statement easily from this: every polynomial with algebraic coefficients may be written as a product of linear terms, each with algebraic integer coefficients, times a rational number.
		Therefore, there exists an integer c_i such that c_ia_i and c_ib_i are algebraic integers (find the one for a_i and the one for b_i, and take c_i to be their lcm).
	www.math.niu.edu /~rusin/known-math/01_incoming/alg_int (462 words)

	Second Test Review (Site not responding. Last check: 2007-11-06)
		Algebraic integers; the algebraic integers form a subring of
		The number of irreducible complex characters is equal to the number of conjugacy classes.
		Since the sum of two algebraic integers is an algebraic integer, we see that
	www.math.vt.edu /people/linnell/5114/Rev2 (326 words)

	Proof: Problem with algebraic integers (Site not responding. Last check: 2007-11-06)
		It says that given two algebraic integers C and b, either C is coprime to b or else it is not coprime to b.
		Over a hundred years ago the ring of algebraic integers became a part of the mathematical lexicon, waiting until now for an intriguing problem with the ring to be revealed by the use of advanced polynomial factorization techniques, which work by using non-polynomial factors of a polynomial.
		Similarly, C is an algebraic integer, so as it is a factor of P(0), if it has a factor in common with b^2, then that factor gets divided off when b^2 is divided off of P(0).
	www.thehelparchive.com /new-2351441-277.html (10722 words)

	Lecture Summary MP473, Number Theory IIIH/IVH
		The algebraic numbers form a field, the algebraic integers form a ring.
		Examples of cyclotomic polynomials, Heinz Lüneberg's account of cyclotomic polynomials, culminating in an algorithm for calculating the mth cyclotomic polynomial which is used in CMATR
		th power of an ideal is principal, gcd's exist in the ring of all algebraic integers.
	www.numbertheory.org /courses/MP473/lectures.html (849 words)

	Some math, algebraic integers (Site not responding. Last check: 2007-11-06)
		By contrast, and algebraic integer is a complex number which is the root of a monic polynomial with INTEGER coefficients.
		At any rate, it's evident that there are algebraic integer factors in common with q_1 and 2, and that those factors cannot be units in the ring of algebraic integers.
		With algebraic integers, it's a little more complicated as if x and y are irrational, and, get this, are roots of a non-monic quadratic with integer coefficients irreducible over rationals, then apparently both x and y must have factors in common with 2.
	www.thehelparchive.com /new-2305688-278.html (19533 words)

	► » New paper, algebraic integers, Galois Theory (Site not responding. Last check: 2007-11-06)
		exists an algebraic integer that is the root of a non-monic polynomial
		of algebraic integers, contrary to the claim of your paper.
		integer", and by substituting for "x,y are coprime
	www.science-chat.org /New-paper-algebraic-integers-Galois-Theory-6894052.html (3236 words)

	LMS Proceedings Abstract, paper PLMS 1428 (Site not responding. Last check: 2007-11-06)
		In 1969, H. Davenport and W. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3.
		In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers $\xi$.
		We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3.
	www.lms.ac.uk /publications/proceedings/abstracts/p1428a.html (139 words)

	Classification error, algebraic integer issue
		be written as the ratio of algebraic integers.
		algebraic integer, as that's necessary for that polynomial to be
		algebraic integers as roots of monic polynomials with integer
	www.groupsrv.com /science/post-7245.html (5581 words)

	Math 662.602, Fall 2005 (Papanikolas)
		This course will be an introduction to the study of algebraic numbers and algebraic integers.
		Of course in number theory the key motivating problem is to understand the basic arithmetic of the integers.
		Algebraic number theory is the study of generalizations of integers to other domains, especially to number fields, ie, finite algebraic extensions of Q. Interesting problems arise in the study of rings of algebraic integers that shed light on many basic number theory problems.
	www.math.tamu.edu /~map/courses/662-fa05 (174 words)

	[No title]
		Then there exists a number field K, with ring of integers R, such that f(x) can be wrirtten as: f(x) = c_f f(x) where c_f is a constant, and f is a primitive polynomial with algebraic integer coefficients.
		If it can be factored into a product of polynomials with algebraic coefficients (not necessarily algebraic integers), then it can be factored into a product of polynomials of the same degree with algebraic integer coefficients.
		QED COROLLARY: Every polynomial with integer coefficients may be factored into a product of linear factors, each with algebraic integer coefficients.
	www.math.niu.edu /~rusin/known-math/01_incoming/alg_factor (1049 words)

	Fermat's Last Theorem for Quadratic Integers (Site not responding. Last check: 2007-11-06)
		I don't know the answer to this question, but there are many solutions in the form A + B sqrt(d) for various rational (non- square) values of d.
		To help decide whether or not solutions exist for a particular exponent p (but not for the general case), it turns out that Kummer's methods of 1850 can be applied to arbitrary rings of algebraic integers (cf.
		(These are analogous to the "regular primes" in Kummer's theorem on FLT for rational integers, but of course we need to determine the set of regular primes for the particular algebraic ring in question.) I don't know if Wiles' general proof of FLT for rational integers covers any of these other algebraic rings.
	www.mathpages.com /home/kmath233.htm (236 words)

	Science: Mathematics: Number Theory: Algebraic - Open Site
		An element of an algebraic number field: an element, a, of a field containing Q which satisfies a polynomial equation with rational coefficients.
		The algebraic integers in an algebraic number field are closed under addition and multiplication, so form a ring, the maximal order.
		The units of the ring of integers of a number field of degree n and signature n=r+2s form a finitely generated abelian group with a torsion-free component of rank r+s-1.
	open-site.org /Science/Mathematics/Number_Theory/Algebraic (467 words)

	Ninth Homework Solutions (Site not responding. Last check: 2007-11-06)
		Since the algebraic integers form a subring, we deduce that
		is an algebraic integer, which is not true because rational algebraic integers are integers.
		Let p be a prime, let q be a positive integer prime to p, let G be a nonabelian simple group of order pq, and let g
	www.math.vt.edu /people/linnell/5114/Ahw9 (410 words)

	Algebraic number (Site not responding. Last check: 2007-11-06)
		All rational numbers are algebraic because every fraction a / b is a solution of bx - a = 0.
		If an algebraic number satisifies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number of degree n.
		All numbers which can be written starting from the rationals using only the arithmetical operations +,-,,/ and square roots, cube roots etc. are algebraic*.
	www.enlightenweb.net /a/al/algebraic_number.html (345 words)

	Algebraic Integers
		algebraic infants, algebraic geometry, algebraic geometry weblog, algebraic graphs, algebraic group, algebraic group theory, algebraic horizontal axis test, algebraic inequality, algebraic inequality in standard form equivalent,
		Many algebra tutorials teach you how to solve math equations that barely look like what you're trying to do.
		I need help with algebraic integers and the math labs are no help at all and my friends are no help.
	www.algebra-answer.com /algebra-helper/algebraic-integers.html (486 words)

	Frequency Sampling Filters With Algebraic Integers (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: Algebraic integers have been proven beneficial to DFT and nonrecursive FIR filter designs [2, 4] since algebraic integers can be dense in C, resulting in short word width, high speed designs.
		This paper uses another property of algebraic integers: algebraic integers can produceexact pole zero cancellation pairs that are used in recursive FIR, frequency sampling filter designs.
		INTRODUCTION An element of C is an algebraic integer if it is a zero of a monic polynomial in F [x] where F is one...
	citeseer.ist.psu.edu /8396.html (363 words)

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