midtermsols2 (1)

midtermsols2 (1) - Math 5c Midterm Due Wednesday, April 28,...

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Unformatted text preview: Math 5c Midterm Due Wednesday, April 28, 4 pm Do the following 5 problems in 3 hours. You may cite any theorems proven in class or Dummit and Foote, but prove any material you use from other sources. You may use technology for calculations. 1. Let E and K be finite extensions of the field F , both contained in a larger field L . If both E and K are splitting fields over F , prove that (a) The composite field EK is a splitting field over F . Let E be the splitting field of f ∈ F [ x ], and K the splitting field of g ∈ F [ x ]. By definition, EK is the smallest field that contains the roots of both f and g , and thus is the splitting field of the polynomial h = fg . (b) The intersection E ∩ K is a splitting field over F . E ∩ K is a normal field for the family of polynomials { f ∈ F [ x ] | all roots of f ∈ E and ∈ K } . Since E ∩ K is a finite extension, it is the splitting field of some polynomial....
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This note was uploaded on 07/19/2010 for the course MA 5c taught by Professor Susamaagarwala during the Spring '09 term at Caltech.

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midtermsols2 (1) - Math 5c Midterm Due Wednesday, April 28,...

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