REVIEW SHEET FOR NEW MATERIAL FOR FINAL

• algebraic/transcendental elements over a field
• minimum polynomial (equivalently irreducible polynomial)
• degree of an algebraic element over a field
• linear independence of vectors in a vector space
• basis of a vector space
• algebraic extension of fields
• finite extension of fields
• degree of a finite extension of fields
• an algebraically closed field
• an automophism of a field
• an automorphism of a field K fixing a subfield F
• an algebraic closure of a field
• a splitting field of a family of polynomials
• multiplicity of a zero of a polynomial
• an embedding of fields
• an embedding of fields over F
• a normal extension
• a separable extension
• a Galois extension

• the minimum polynomial is irreducible
• F(alpha) is isomorphic to F[x]/(irr(alpha,F))
• finite extensions are algebraic
• K/Q a field extension and phi an automorphism of K implies that phi fixes Q.
• K/F algebraic and phi an automorphism of K which fixes F. Then phi(alpha) is a zero of irr(alpha,F).
• F is a characteristic 0 field, alpha is a zero of g(x) with multiplicity e, then alpha is a zero of g'(x) with multiplicity e-1.
• If K is a splitting field over F, then every embedding tau of K into F over F satisfies tau(K) is contained in K.
• If L/K/F are a tower of fields and L/F is separable, then L/K and K/F are separable.
• If E is a subfield of a Galois extension K/F, then K^Gal(K/E) = E.