The axiom of choice indicates that each of these mathematical objects has an algebraic closure which contains it, and For a polynomial over one of these, the "splitting" one is the smallest extension to one of these that allows the polynomial to factor into linear factors.
The axiom of choice indicates that each of these mathematical objects has an algebraic closure which contains it, and For a polynomial over one of these, the "splitting" one is the smallest extension to one of these that allows the polynomial to factor into linear factors. Finite ones must have order equal to a power of a prime, and the fact that the constructible numbers form one of these structures is used to prove that (*) doubling a cube and trisecting an angle are impossible. These objects by definition have a Krull dimension of zero, and the fact that polynomials of degree higher than four have no closed solutions is proved using extensions of these objects, which are the subject of Galois theory. These objects are closed, associative, and commutative over both addition and multiplications, and are thus less general than rings. For 10 points, name these algebraic structures.
The axiom of choice indicates that each of these mathematical objects has an algebraic closure which contains it, and For a polynomial over one of these, the "splitting" one is the smallest extension to one of these that allows the polynomial to factor into linear factors.
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