Finite dimensional extension field

Related BrainMass Solutions

• Algebraic Extensions

  obtained by adjoining u to F Theorem 10.9 If K is a finite-dimensional extension field of F,then K is an algebraic extension of F.

• Splitting Fields : Let F be an extension field of K of degree 2, then F is the splitting field over K for some polynomial.

  Proof: Theorem: If F is a finite dimensional extension field of K, then F is finitely generated and algebraic over K. According to this theorem, since [F:K]=2 is finite, then F is algebraic over K.

• Finite Extension Field and Isomorphism

  39353 Finite Extension Field and Isomorphism Argue that every finite extenstion field of R is either R itself or is isomorphic to C.

• Basis and Finite and Infinite Field Extensions

  58988 Basis and Finite and Infinite Field Extensions Find a basis for the extension of and also calculate We know already that is infinite.

• Hahn Banach Theorem Application

  Prove that each finite dimensional subspace of is complemented in . Hint: Suppose that M is a finite dimensional subspace of and is a basis for M. Define linear functionals : M K by for in K, k=1,2,...,n.

• Galois Theory : Use the method found in the proof below to find a primitive element for the extension Q(i, 5^(1/4)) over Q.

  If F is separable over K, then it is a simple extension of K. Proof: Let F be a finite separable extension of K. If K is a finite field, then F is a finite field.

• Show that a finite extension of fields K/k which have prime degree have only trivial subextensions.

  80989 finite extension of fields K/k which have prime degree Show that a finite extension of fields K/k which have prime degree have only trivial subextensions.

• Splitting field proofs

  the normality criterion ensures that K1 n K2 is a * finite * normal extension, hence a splitting field over F This provides two splitting field proofs.

• Embeddings

  28123 Finite Set of Embeddings Let E be a finite extension of a field F. Show that any finite set of distinct embeddings of E into the algebraic closure of F is linearly independent over F. See attached pdf file.