## Math 342 study

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mworley88  on March 4, 2011

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# Math 342 study

 splitting field for the polynomial f(x)A field in which f(x) factors completely into linear factors and f(x) dies not factor into completely into linear factors in a subfield of the splitting field
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#### Definitions

splitting field for the polynomial f(x) A field in which f(x) factors completely into linear factors and f(x) dies not factor into completely into linear factors in a subfield of the splitting field
normal extension of F an algebraic extension of F which is the splitting field over F of a collection of polynomials f(x) contained in F
primitive nth root of unity Generator of the cyclic group of all nth roots of unity
cyclotomic field of nth roots of unity Q(zeta_n)
Theorem 27 if f(x) is contained in F[x] is a polynomial and f'(x) contained in F'[x] is the polynomial applying tsi to the coefficients of f(x). Let E be a splitting field for f(x) over F and E' be a splitting field for f'(x) over f'(x), Then the isomprphism extends to E
algebraic closure of F a field that is algebraic over F and every polynomial in F splits completely over the field.
algebraically closed every polynomial with coefficients in K has a root in K.
derivative of a polynomial f(x)=sum (p_ix^(p_i-1)) for all p_i in (0,deg f) (the usual derivative)
(a+b)^p in a field of characteristic p a^p+b^p
perfect field of characteristic P field in which every element of K is a pth power in K, i.e., K=K^p. If a field has characteristic 0, it is also called perfect
Frobenius Endomorphism phi(a)=a^p, an injective homomorphism.
Fundamental Theorem of Algebra The Field C is algebraically closed
K is separable over F if every element of K is the root of a separable polynomial over F.
mu_n group of nth roots of unity over n
cyclotomic polynomial (elephant trunk) polynomial whose roots are the primitive nth roots of unity.
degree of cyclotomic polynomial Euler phi function

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