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GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3

Ga2 Local Eisenstein UniformizerGa2 Different Exponent Tame

Separable, normal, and algebraic extensionsGraduate Algebra II · Mixed review

GA:020118In Q(i) the prime 2 ramifies as (1+i)^2, 3 is inert with residue F_9, and 5 splits, with Frob_3 conjugation

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Ga2 Gaussian Splitting TypesGa2 Frobenius Quadratic Imaginary

Separable, normal, and algebraic extensionsGraduate Algebra II · Mixed review

GA:020219The extension Q(zeta_5)/Q is cyclic of degree 4 with unique quadratic Q(sqrt(5)), trace -1, and discriminant 125

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Ga2 Fifth Cyclotomic Galois LatticeGa2 Real Subfield Quadratic

Finite and introductory infinite Galois theoryGraduate Algebra II · Mixed review

GA:020227Compatible 2-adic partial sums of (-2)^k invert 3 with residues 1,3,3,11,11,43 though the real series diverges

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Ga2 Two Adic Inverse ThreeGa2 Compatible Inverse System Z2

Finite and introductory infinite Galois theoryGraduate Algebra II · Mixed review

GA:020320The cusp k[t^2,t^3] is not normal, has conductor t^2 k[t] and delta 1, and is singular of tangent dimension 2

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Ga2 Cusp NormalizationGa2 Conductor Delta Invariant

Noetherian rings and primary structureGraduate Algebra II · Mixed review

GA:020421The local ring k[x,y]_(x,y) has Hilbert-Samuel length n(n+1)/2, associated graded k[x,y], and multiplicity 1

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Ga2 Hilbert Samuel Regular PlaneGa2 Associated Graded Multiplicity

Localization, dimension, and local algebraGraduate Algebra II · Mixed review

GA:020422In k[x,y]_(x,y) the pair (y),(y-x^2) has colength 2 on the basis 1,x while the transverse line x=0 has length 1

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Ga2 Tangent Intersection LengthGa2 Local Colength Basis

Localization, dimension, and local algebraGraduate Algebra II · Mixed review

GA:020428Localizing the quadric cone k[x,y,z]/(xz-y^2) at (x,y) is the DVR k(z)[y]_(y) with v(x)=2

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Ga2 Quadric Cone Height One DvrGa2 Localized Uniformizer Valuations

Localization, dimension, and local algebraGraduate Algebra II · Mixed review

GA:020523Koszul homology of (x,xy) on k[x,y] is H0=R/(x), H1=R/(x), H2=0, so the sequence is not regular

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Ga2 Nonregular Koszul HomologyGa2 Koszul Cycle Boundary

Ext, Tor, and derived constructionsGraduate Algebra II · Mixed review

GA:020524The Cech tail H^1_(x)(k[x])=R_x/R has basis x^{-1},x^{-2},... is not finitely generated, and x kills the first pole

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Ga2 One Variable Local CohomologyGa2 Cech Tail Not Finitely Generated

Ext, Tor, and derived constructionsGraduate Algebra II · Mixed review

GA:020529The A2 quiver has Ext^1(S1,S2)=k nonsplit while Ext^1(S2,S1)=0 because S2 is projective

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Ga2 A2 Quiver Ext AsymmetryGa2 Projective Resolution Simples

Ext, Tor, and derived constructionsGraduate Algebra II · Mixed review

GA:020625The blowup of A^2 at the origin separates the lines y=0, y=x, and x=0 along the exceptional P^1 at [1:0],[1:1],[0:1]

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Ga2 Blowup Separates Three LinesGa2 Strict Transform Exceptional Incidence

Affine varieties and schemes as an interfaceGraduate Algebra II · Mixed review

GA:020626The smooth conic x^2+y^2=z^2 over F_5 is parametrized bijectively by P^1(F_5) onto six explicit points

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Ga2 F5 Projective ConicGa2 P1 Parametrization Finite Field

Affine varieties and schemes as an interfaceGraduate Algebra II · Mixed review

GA:020630The curve y^2=x^3+x+1 over F5 has nine rational points and is cyclic of order 9 generated by (0,1)

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Ga2 F5 Elliptic Nine Cyclic PointsGa2 Weierstrass Group Law Order

Affine varieties and schemes as an interfaceGraduate Algebra II · Mixed review

GA:020631The twisted cubic in P3 has Hilbert polynomial 3n+1, value 7 in degree 2, degree 3 and genus 0

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Ga2 Twisted Cubic Hilbert PolynomialGa2 Veronese Binary Forms

Affine varieties and schemes as an interfaceGraduate Algebra II · Mixed review

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GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3

GA:020118In Q(i) the prime 2 ramifies as (1+i)^2, 3 is inert with residue F_9, and 5 splits, with Frob_3 conjugation

GA:020219The extension Q(zeta_5)/Q is cyclic of degree 4 with unique quadratic Q(sqrt(5)), trace -1, and discriminant 125

GA:020227Compatible 2-adic partial sums of (-2)^k invert 3 with residues 1,3,3,11,11,43 though the real series diverges

GA:020320The cusp k[t^2,t^3] is not normal, has conductor t^2 k[t] and delta 1, and is singular of tangent dimension 2

GA:020421The local ring k[x,y]_(x,y) has Hilbert-Samuel length n(n+1)/2, associated graded k[x,y], and multiplicity 1

GA:020422In k[x,y]_(x,y) the pair (y),(y-x^2) has colength 2 on the basis 1,x while the transverse line x=0 has length 1

GA:020428Localizing the quadric cone k[x,y,z]/(xz-y^2) at (x,y) is the DVR k(z)[y]_(y) with v(x)=2

GA:020523Koszul homology of (x,xy) on k[x,y] is H0=R/(x), H1=R/(x), H2=0, so the sequence is not regular

GA:020524The Cech tail H^1_(x)(k[x])=R_x/R has basis x^{-1},x^{-2},... is not finitely generated, and x kills the first pole

GA:020529The A2 quiver has Ext^1(S1,S2)=k nonsplit while Ext^1(S2,S1)=0 because S2 is projective

GA:020625The blowup of A^2 at the origin separates the lines y=0, y=x, and x=0 along the exceptional P^1 at [1:0],[1:1],[0:1]

GA:020626The smooth conic x^2+y^2=z^2 over F_5 is parametrized bijectively by P^1(F_5) onto six explicit points

GA:020630The curve y^2=x^3+x+1 over F5 has nine rational points and is cyclic of order 9 generated by (0,1)

GA:020631The twisted cubic in P3 has Hilbert polynomial 3n+1, value 7 in degree 2, degree 3 and genus 0

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GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3

GA:020118In Q(i) the prime 2 ramifies as (1+i)^2, 3 is inert with residue F_9, and 5 splits, with Frob_3 conjugation

GA:020219The extension Q(zeta_5)/Q is cyclic of degree 4 with unique quadratic Q(sqrt(5)), trace -1, and discriminant 125

GA:020227Compatible 2-adic partial sums of (-2)^k invert 3 with residues 1,3,3,11,11,43 though the real series diverges

GA:020320The cusp k[t^2,t^3] is not normal, has conductor t^2 k[t] and delta 1, and is singular of tangent dimension 2

GA:020421The local ring k[x,y]_(x,y) has Hilbert-Samuel length n(n+1)/2, associated graded k[x,y], and multiplicity 1

GA:020422In k[x,y]_(x,y) the pair (y),(y-x^2) has colength 2 on the basis 1,x while the transverse line x=0 has length 1

GA:020428Localizing the quadric cone k[x,y,z]/(xz-y^2) at (x,y) is the DVR k(z)[y]_(y) with v(x)=2

GA:020523Koszul homology of (x,xy) on k[x,y] is H0=R/(x), H1=R/(x), H2=0, so the sequence is not regular

GA:020524The Cech tail H^1_(x)(k[x])=R_x/R has basis x^{-1},x^{-2},... is not finitely generated, and x kills the first pole

GA:020529The A2 quiver has Ext^1(S1,S2)=k nonsplit while Ext^1(S2,S1)=0 because S2 is projective

GA:020625The blowup of A^2 at the origin separates the lines y=0, y=x, and x=0 along the exceptional P^1 at [1:0],[1:1],[0:1]

GA:020626The smooth conic x^2+y^2=z^2 over F_5 is parametrized bijectively by P^1(F_5) onto six explicit points

GA:020630The curve y^2=x^3+x+1 over F5 has nine rational points and is cyclic of order 9 generated by (0,1)

GA:020631The twisted cubic in P3 has Hilbert polynomial 3n+1, value 7 in degree 2, degree 3 and genus 0

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GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3View problem statement

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Problem results

GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3View problem statement

GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3

GA:020117X^2-3 is 3-Eisenstein, so Q_3(sqrt(3)) has uniformizer pi=sqrt(3), different (pi), and residue F_3