Let where . Verify that is also a root of . Determined . Show that is normal over .
means . So we only need to prove is another root of .
Let , means .
Let , then , obviously , thus is a root of .
is monic and has no root in , so it is irreducible in . is minimal polynomial of . So . must be the cyclic group of order 3. By theorem 4.7 chapter 4 in Jacoboson’s book, is normal over .
Jacobson, Algebra I. p243.

Search

Recent

Categories
 Algebra (23)
 Galois Theory (20)
 Group Theory (2)
 Rings, Modules, Homology (2)
 Analysis (42)
 Complex Analysis (11)
 Harmonic Analysis (10)
 Inequalities (8)
 Real Analysis (16)
 Geometry (36)
 Hyperbolic Geo (3)
 Riemannian Geo (32)
 PDE (75)
 Elliptic PDE (46)
 Mean curvature flow (10)
 NavierStokes Eqns (9)
 Parabolic PDE (2)
 Schrodinger Eqn (1)
 Yamabe flow (4)
 Uncategorized (14)
 Algebra (23)

Tags
Algebraically closed alternating group barrier function Besicovitch caccioppoli ineq conformally invariant convergence radius 1 covering lemma cubic equation cyclic extension cyclotomic field dihedral group Dirichlet problem discriminant distance function Eisenstein criterion field of characteristic p finite extension fourier transform fully nonlinear galois group Gamma function geodesic green formula green function harmonic functions harnack inequality Hibert's Satz 90 Hilbert transform holder hypersurface Hölder Continuity injective intermediate subfield intersection invariant field irreducible Jacobi field line segment maximum principle Mean Oscillation Minkowski inequality Neumman problem newtonian potential nonnegative coefficients nonextentable poincare ineq pole polynomial primitive element primitive root of unity projective module regularity Riemann mapping Riemann surface root tower rouche separable sigma2 simply connected singular point sobolev space solenoidal vector field solvable group splitting field subgroup symmetric group trace and norm transitive uniform continuous uniqueness continuation univalent weak derivative yamabe zero 
Blog Stats
 39,492 hits

Meta