SUMS OF TWO SQUARES 1. Sums of two squares We would now like to ...
by PL CLARK - Cited by 2 http://www.math.uga.edu/~pete/4400twosquares.pdf
QUADRATIC RINGS 1. Quadratic fields and quadratic rings Let D be a ...
c) A prime p is of the ofrm |x2 − 3y2| iff 3 is a square modulo p. ... if p ≡ −1 (mod 4) is a prime number, then it remains irreducible in Z[ ... http://math.uga.edu/~pete/4400quadrings.pdf
^I (6)
Let hereafter p be a prime = 3. (mod 4), and denote by T, U the fundamental ... (.p) = PP-1, Further since p remains a prime in the quadratic field k(i), the irreducible equation \-\-z-\-z2-\-...-\-zv~x = 0 satisfied by I is still ... http://jlms.oxfordjournals.org/cgi/reprint/s1-36/1/282.pdf
Rings & Arithmetic 9: The Gaussian integers
If p ≡ 3 (mod 4) then p remains prime in Z[i]. • If p ≡ 1 (mod 4) then p becomes reducible in Z[i]—in fact p factorises as a product of two distinct ... http://people.maths.ox.ac.uk/~flynn/genus2/alg0506/RandALect9.pdf
MATH 123: ABSTRACT ALGEBRA II SOLUTION SET # 4 1. Chapter 11 ...
If p = 11,x2 + 3 ≡ 0 mod 11 is irreducible by inspecting 12 through 62. Thus ... Problem 3 (a) Suppose that a prime integer p remains prime in R. Prove that ... such that y is the remainder when we set x ≡ a and z is the remainder when we ... Problem 7 Assume that d ≡ 2 or 3 mod 4. Prove that a prime integer p ... http://www.math.harvard.edu/~ctm/home/text/class/harvard/123/02/html/home/solns/sset-4.pdf
Sums of Two Squares
If π is an irreducible (prime) element of Z[i] and α, β ∈ Z[i] then π|αβ ... If p ≡ 3 mod 4 you showed on the midterm that p is not a sum of two squares. ... The only possibility that remains is N(π)= p, so p =a2 + b2. ... http://sporadic.stanford.edu/bump/sumsofsquares.pdf
On the extraspecial case of the k(GV)-conjecture
1 and p be an (odd) prime with p ≡ 3 (mod 4). Since E is absolutely irreducible on V, ..... But each of the characters θ2,θ3 and θ5 remains irreducible mod 3 ... NK (T) and NK (T) = Z(E)(P Ũ Y) where either Y = T or Y is isomorphic to ... http://www.springerlink.com/index/AH3A77PGVXT19DUP.pdf
Efficient pth root computations in finite fields of characteristic p
irreducible binomials over Fq, where q is a power of an odd prime p, p ≥ 5. Then we ... (2) each prime factor of m divides e,. (3) if m ≡ 0 (mod 4) then q ≡ 1 (mod 4). ..... What remains is to precompute z1/p, z2/p,..., z(p−1)/p ... http://www.springerlink.com/index/l0866w5183046555.pdf
Irreducible polynomials and Barker sequences
by P Borwein - Cited by 3 http://www.davidson.edu/math/mossinghoff/IrredPolyBarker_BKM.pdf
Primes of the form x + ny where n ≡ 3 mod 4
by L Elliott - Related articles http://arts.uwaterloo.ca/~ltelliot/primes-of-the-form-project.pdf
Discrete Math, Fourteenth Problem Set (July 18)
irreducible. Indeed, we have a deterministic test for irreducibility of .... but 3 remains a prime. Exercise 0.10. For z = a + bi, let N(z) = a2 + b2 (the ... Prove that if p is a prime number (in Z) and p ≡ −1 (mod 4) then p is a ... http://people.cs.uchicago.edu/~laci/reu03/notes14.pdf
Lecture 19 (Primes, Maximal Ideals and Fields)
It remains to show the non zero elements of R/πR have (multiplicative) inverses. ... Recall if p ≡ 3 mod 4 is prime number, p is irreducible in Z[i]. So ... http://talus.maths.usyd.edu.au/u/bobh/UoS/MATH3962/r/lecture19.pdf
Irreducible polynomials and Barker sequences
by P Borwein - Cited by 3 http://www.math.ncsu.edu/~kaltofen/bibliography/07/BKM07.pdf
Irreducible disjoint covering systems
by I Koreosystem (DCS) of congruence classes on Z. It will allow us to decompose every natural DCS to some full systems Zp, p a prime, of congruence classes modulo p. .... For example, Z4 is reducible, because 0(mod 4) u 2 (mod 4) = 0(mod 2) .... It remains to prove p"* J M. Consider arbitrary jV i, 1 < _/ < k — 1 (the ... http://matwbn.icm.edu.pl/ksiazki/aa/aa44/aa4448.pdf
Irreducibility of the iterates of a quadratic polynomial over a field
by M Ayad - 2000 - Cited by 12 http://matwbn.icm.edu.pl/ksiazki/aa/aa93/aa9317.pdf
Factoring of Prime Ideals in Extensions
linear factors, hence x2 − m is irreducible mod p and p remains prime. ... If m ≡ 3 or 7 mod 8, then m ≡ 3 mod 4, so there is only one remaining case. ... 2), and assume as known that the ring of algebraic integers is B = Z[ 3 ... http://www.math.uiuc.edu/~r-ash/Ant/AntChapter4.pdf
ALGEBRAIC NUMBER THEORY (1) Find the fundamental unit of Q( √ 6 ...
ization in Z[. √. −10 ], and show also how this factorization can be explained ... We first have to show that the factors are irreducible and do not differ by ... (3) Let p ≡ 3 mod 4 be prime. Show that exactly one of the two equations ... It remains to show that at most one of them has a solution in integers. ... http://www.fen.bilkent.edu.tr/~franz/ant/m1a.pdf
Gauss Units in Integral Group Rings
by O Neisse - 1998 - Cited by 2 http://www.math.ualberta.ca/people/Faculty/Sehgal/publications/085.pdf
Arithmetic in the ring of formal power series with integer ...
by D Birmajer - Cited by 6 http://math.aa.psu.edu/~juan/papers/ufdPower.pdf
Previous exams
(b) Deduce that p is irreducible in Z[i] if and only if there is no ... (a) Prove that if n ≡ 3(mod 4), then there are no integer solutions ... (b) Deduce that any prime number p which is congruent to 3 modulo. 4 remains prime in Z[i]. ... http://www.unizar.es/matematicas/algebra/elduque/IntroAlgebra/IAPrevExam.pdf
ARTICLES
and n, m + ni is irreducible in Z[¿] of norm p. (ii) For a rational prime p = 3(mod4), p remains irreducible in Z[i] of norm p2. ... http://mathdl.maa.org/images/upload_library/22/Allendoerfer/1997/0025570x.di021195.02p0087x.pdf
Some analogue of Hilhert's irreducibility theorem and the ...
- Related articles http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/1759/1/jfs380104.pdf
MATH 632 Homework List 4 27 March 2000 Sections 4, 5, & 6 ...
Another way to state the result is: every prime integer p ≡ 3 (mod 4) remains ir- reducible in Z[i], while every other prime integer p factors as p = qq, ... https://pantherfile.uwm.edu/adbell/www/Teaching/632/2000/hmwk4.pdf
Finding Isomorphisms Between Finite Fields Author(s): H. W. ...
by HW LENSTRA JR - 1991 - Cited by 50 http://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1991b/art.pdf
C:/Documents and Settings/c_flaut/My Documents/ANALE-2005/ARTICOL ...
by D Savin - Related articles http://www.emis.de/journals/ASUO/mathematics/pdf7/dsavin_81_88.pdf
NORMSETS AND DETERMINATION OF UNIQUE FACTORIZATION IN RINGS OF ...
- Related articles http://www.mrlonline.org/proc/1996-124-06/S0002-9939-96-03261-3/S0002-9939-96-03261-3.pdf
THE CUBIC CONGRUENCE x3 + Ax2 + Bx + C = O(modp) AND BINARY ...
by BK SPEARMAN - 1992 - Cited by 9 http://zakuski.math.utsa.edu/~jagy/Spearman_Williams_1992.pdf
CONSTRUCTION OF POLYNOMIALS IRREDUCIBLE OVER A FINITE FIELD WITH ...
a power of a prime p. The characteristic polynomial of X is an irreducible polynomial ..... and q = 3 (mod 4). If 2y is the greatest power of 2 dividing n, then we set v(n) = γ. ... where b = a -
Irreducible Quartic Polynomials with Factorizations modulo p
s − r ≡ 3 (mod 4). Hence, by Theorem 5, the polynomial Lk(x) is irreducible in Z[x] but reducible mod- ulo p for every prime p independently of k. ... http://www.math.carleton.ca/~williams/papers/pdf/281.pdf
ABOUT THE PERIOD OF BELL NUMBERS MODULO A PRIME Reprinted from the ...
by M Car - 2008 - Cited by 3 http://icms.kaist.ac.kr/mathnet/thesis_file/17_B07-0730.pdf
Notes on Quartic Fields
by Y Motoda - Cited by 5 http://dlwww.dl.saga-u.ac.jp/contents/mgzn/ZR00004546/ZR00004546.pdf
Algebra for Digital Communication
Hence it is isomorphic to Z/15Z, and the number of elements of ... Then X2 − a ∈ Fp[X] is either irreducible or has exactly two ... It remains to show, that there exists an isomorphism σ : F10609 → F10609 such that σ(α) = −α. ... Let p be any prime number with p ≡ 3 (mod 4). Then we know by exercise ... http://alg-geo.epfl.ch/cours/syscom/exo0607/solutions11.pdf
COPRIMENESS AMONG IRREDUCIBLE CHARACTER DEGREES OF FINITE SOLVABLE ...
to be maximal such that G/K remains nonabelian. Notice that, in this situation ... of G/K. Now since G has no nonabelian p-factor groups for any prime p, the .... Recall that an arbitrary member of s(z) has the form rz, where ... q ≡ 3 mod 4, we can show that these are the only irreducible character degrees of ... http://www.ams.org/proc/1997-125-10/S0002-9939-97-04269-X/S0002-9939-97-04269-X.pdf
ß,(x)-Z%-(x>-l-*-i).
Let fp(X) denote the irreducible polynomial of Hp over Q. A table of these .... 3. Bernoulli numbers and Fermât quotients. Let p be an odd prime. For x an .... for which the theorem remains valid if we replace f (X) in (b) by Irr(a, L/Q, X). If .... (b) If p = 1 (mod4), then 1(a) > p for every a G 0 — Z. Remark. ... http://www.ams.org/tran/1985-288-01/S0002-9947-1985-0773047-X/S0002-9947-1985-0773047-X.pdf
(R(y)f)(x) f(xy), f L2(G(Q)\G(A)), x,y G(A),
by J Arthur - 1997 - Related articles http://claymath.org/cw/arthur/pdf/45.pdf
Galois actions on preimage trees
P(Of (a)) = {p prime : p divides some element of Of (a)}. (Can extend to rational functions .... remains unknown, but appears to have infinite index in Aut(T∞ ... All p in S are ≡ 1 (mod 4). Rafe Jones. Galois actions on preimage trees .... Call h ∈ Z/pZ[x] f -stable if h ◦ f m is irreducible in Z/pZ[x] for all ... http://mathcs.holycross.edu/~rjones/MPI3_singlepages.pdf
Lecture # 8 and 9: Ideals and the Class Number Formula.
p if p ≡ 3 (mod 4) π¯π (with π = u¯π) if p ≡ 3 (mod 4) .... An ideal A is irreducible if and only if it is maximal (that is not properly contained ... Now we can argue just was we did in Z[i] and a prime p factors in O√ .... All that remains to show is that this correspondence preserves the equivalence class ... http://math.berkeley.edu/~nsnyder/tutorial/lecture89.pdf
AND A THEOREM OF VON NEUMANN
by RJ SPATZIER - 1989 - Cited by 12 http://www.math.lsa.umich.edu/~spatzier/isospectral.pdf
P. ErdËos, On integers of the form 2 related problems, Summa ...
x ≡ 1 (mod 4) x ≡ 3 (mod 8) x ≡ 7 (mod 12) x ≡ 23 (mod 24). P. ... expressible in the form a + p with a ∈ A and p prime? ..... an f(x) ∈ Z[x] satisfying f(x)xn + 1 is reducible ... digit, then the number remains composite. Are ... http://www.math.sc.edu/~filaseta/seminars/OpenCoveringProblems2009.pdf
IRREDUCIBILITY CRITERIA OF SCHUR-TYPE AND P´OLYA-TYPE 1 ...
by K GYORYproved that a polynomial P(x) ∈ Z[x] of odd degree m is irreducible if ... m > 5,d> 0,d ≡ 3 mod 4. In 1933 Dorwart and Ore [12] generalized various of the above men- .... In case c is a prime p, Dorwart and Ore [12], Theorem 14, ...... and (n − k − 1)!, respectively, and the statement remains valid. ... http://www.hausdorff-research-institute.uni-bonn.de/files/preprints/diophantine/ght15.pdf
Optimal Rate-Diversity Tradeoff STBCs from Codes over Arbitrary ...
d] if d ≡ 2,3 mod4 and Z[ .... irreducible polynomial for constructing F172 over F172 and the ... Example 3: Let p = 3, which remains a prime in Z[i]. ... http://ieeexplore.ieee.org/iel5/9996/32109/01494393.pdf?arnumber=1494393
BASIC RING THEORY Contents 1. Definitions and examples We are ...
by JK VERMA - Related articles http://www.math.iitb.ac.in/atm/atmt1/jkv.pdf
RESIDUE CLASSES FREE OF VALUES OF EULER'S FUNCTION Kevin Ford ...
by K Ford - Cited by 10 http://www.math.dartmouth.edu/~carlp/phires.pdf
STOR
by RJ Stroeker - 1989 - Cited by 3 http://www.stroeker.nl/MathComp89.pdf
ON THE EXISTENCE OF DISTORTION MAPS ON ORDINARY ELLIPTIC CURVES 1 ...
by D CHARLES - Cited by 4 http://eprint.iacr.org/2006/128.pdf
Decomposition of Prime Ideals in the Extensions
p ≡ 1 (mod 4), then p = x2 +y2 has solutions but if p ≡ 3 (mod 4), ..... p remains prime in. Z[√2]. Ker(Ar) = {(a/b)Z : a and b are odd integers and the ... http://faculty.uaeu.ac.ae/hakca/papers/kendirli.pdf
Counterexamples to the Hasse principle
by M Bright - Related articles http://www.warwick.ac.uk/~maseap/arith/notes/elementary.pdf
Untitled
under the finite symplectic group Sp2n(p), where p 3( mod 4) is prime and n is .... certain cases when p 3(mod4), but again the case p 1(mod4) remains a mystery. ..... is an irreducible (R=rR)ËœG-module for all primes r 6= p. ..... Z-module automorphism of any -stable lattice. If g 2 G+ n H and g ... http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.4dummigan.pdf
WHY FACTORIZATION THEORY: THE BASICS
factorization of α remains the same. This is reflected in the class number by this classical result of .... We call x prime if whenever x | yz, for y and z in M, then either x | y or x | z. .... then either x = p where p is a prime number with p ≡ 1 ... are prime numbers with qj ≡ 3 (mod 4), then any irreducible ... http://www.shsu.edu/~stc008/Keynote.pdf
A Polynomial with Galois Group SL2(11)
by JUKL UNERS - Related articles http://www2.math.uni-paderborn.de/fileadmin/Mathematik/AG-Klueners/publications/sl2_11.pdf
0 komentar:
Posting Komentar