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Unformatted text preview: Math 4108 midterm 2 study sheet April 20, 2010 Know the fact that if f ( x ) F [ x ], where F is a field, and f ( x ) has a double root, then f ( x ) and f ( x ) have a common factor. Know that this is if and only if in fields of characteristic 0 (i.e. in ch. 0, f has a double root iff ( f,f ) negationslash = 1), and the same is true in characteristic p ... but when you go to apply this to irreducible polynomials, something can go wrong: There are fields of characteristic p for which there are irreducbile polynomials f ( x ) satisfying ( f,f ) &gt; 1 irreducible does not always imply that there are no double roots. Know the definition of Galois group, and Galois extension (normal and separable), know what separability means. Know the Fundamental Theorem of Galois theory the one-to-one correspondence between subgroups of the Galois group and subfields, along with the one-to- one correspondence between normal subgroups of the Galois group and normal extensions. It is good to have at least some rough idea of how this is proved. Also, know what solvable by radicals means. Given symbols x 1 ,...,x n (which one can think of as roots of some ab- stract polynomial), let S denote the field of all symmetric rational func- tions f ( x 1 ,...,x n ) /g ( x 1 ,...,x n ) E := F ( x 1 ,...,x n ), where F is some field. Know how to show that E is a splitting field for a certain degree- n polynomial, which therefore implies [...
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