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16 terms
Terms | 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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