## Algebra |

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Abelian group adjunction algebraic extension algebraic number algebraically closed algebraically independent arbitrary automorphism base field basis belongs called coefficients commutative complex numbers composition series conjugate contains corresponds cosets covectors cyclic group defined degree denote divisible equal equation equivalent example Exercises exists expressed extension field field Q finite number following theorem formally real field fundamental sequence Fundamental Theorem Galois group given hence homomorphism identity implies indeterminate induction infinite integral domain inverse irreducible isomorphism least upper bound Lemma linear factors linearly independent mapping matrix modulo multiplication natural numbers nomial normal divisor normal series nth roots number field number of elements obtain one-to-one permutation polynomial f(x prime field prime number principal ideal Proof quotient rational numbers real closed field real numbers residue class residue class ring roots of unity Section splitting field subfield subgroup subset substitutions symmetric functions symmetric group tensor theory transformation vector space