A Course in Metric Geometry Dmitri Burago Yuri Burago Sergei Ivanov

Description

• cours - matière potentielle : description
• exposé

A Course in Metric Geometry Dmitri Burago Yuri Burago Sergei Ivanov Department of Mathematics, Pennsylvania State University E-mail address: Steklov Institute for Mathematics at St. Petersburg E-mail address: Steklov Institute for Mathematics at St. Petersburg E-mail address:

• range of noticeable applications of geometrical technique
• metric geometry
• geometrical topics
• metrics
• group theory
• curvature
• length
• time to time
• time after time
• time time

Sujets

• Description
• Exposé
• Burago
• Ivanov
• Steklov Institute of Mathematics
• Aleksandrov
• Gromov
• Pennsylvania State University
• Cambridge University

Informations

• Publié par: edwui
• Nombre de lectures: 32
• Langue: English

Extrait

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Stanford University Math 121: Abstract Algebra II Lectures

YongwhanLim

Winter 2007-2008

Lecture 1 (Tuesday, January 8)

Historical remarks (solvability of polynomial equations, impossibility of certain compass and straightedge constructions)

Proposition 1(Gauss’s Lemma (page 303)).LetRbe a Unique Factorization Domain with field of fractionsFand letp(x)∈R[x]. Ifp(x)is reducible inF[x]thenp(x)is reducible in R[x].

Proposition 2(Eisenstein’s Criterion (page 310)).Letpbe a prime inZand letf(x) = n n−1 2 a n x+a n−1 x+∙ ∙ ∙+a 0 ∈Z[x],n≥1. Supposepdoes not dividea n andpdoes not dividea 0 butpdividesa 0 ,∙ ∙ ∙, a n−1 . Thenf(x)is irreducible in bothZ[x]andQ[x]. Proposition 3.The polynomialp(x)∈F[x]hasx−aas a factor iffp(a) = 0.

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Lecture 2 (Thursday, January 10)

Definition 1.IfKis a field containing the subfieldFthenKis said to be an extension field ofF, denotedK/F.

Definition 2.The degree of a field extensionK/F, denoted[K:F], is the dimension ofK as a vector space overF.

Definition 3.The elementα∈Kis said to be algebraic overFifαis a root of some nonzero polynomialf(x)∈F[x]. Ifαis not algebraic overFthenαis said to be transcendental over Fextension. The K/Fis said to be algebraic if every element ofKis algebraic overF.

Proposition 4.Letαbe algebraic overFthere is a unique monic irreducible poly-. Then nomialm α,F (x)∈F[x]which hasαas a root. A polynomialf(x)∈F[x]hasαas a root iffm α,F (x)dividesf(x)inF[x]. The polynomialm α,F (x)is called the minimal polynomial forαoverF. The degree ofm α (x)is a degree ofαif. Furthermore, L/Fis an extension of fields andαis algebraic over bothFandLthenm α,L (x)dividesm α,F (x)inL[x].

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