## Lectures on modern higher algebra: Galois theory. Notes by Albert A. Blank, New York university, summer 1947, Part 1 |

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### Contents

Rings and Fields | 15 |

equations in a field 19 Vector spaces | 22 |

Factorization into Primes | 28 |

2 other sections not shown

### Common terms and phrases

3-cycle addition and multiplication algebraic Assume automorphisms clearly coefficients congruence relation Consequently Consider consists construct Corollary corresponding cosets cyclic group defined degree denote digit distinct division elements of F equivalent example Exercise extension field factor group factorization into primes field F field of f(x finite number fixed field follows Furthermore G contains greatest common divisor ground field group G group of order Hence identity images invariant subgroup inverse irreducible equation irreducible polynomial isomorphic Lemma linear combination linear factors linearly independent mapping n-th root nomial non-trivial non-zero obtain p-th permutation polynomial f(x possible postulates primitive principal ideal ring Proof proved radicals rational functions rational numbers real numbers remain fixed residue classes result ring roots of f(x roots of unity satisfies separable polynomial solution solvable splitting field symmetric group Theorem unique factorization vector space whence zero