Abstract
Recall that a field F is a set with two binary operations, addition, denoted by +, and multiplication denoted by. or just by juxtaposition, defined on it satisfying the following nine axioms:
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(1)
Addition is commutative: a +b = b + a for each pair a, b in F.
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(2)
Addition is associative: a + (b + c) = (a + b) + c for a, b, c ∈ F.
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(3)
There exists an additive identity, denoted by 0, such that a + 0 = a for each a ∈ F.
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(4)
For each a E F there exists an additive inverse denoted —a, such that a + (-a) = 0.
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(5)
Multiplication is associative: a(bc) = (ab)c for a, b, c ∈ F.
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(6)
Multiplication is distributive over addition: a(b + c) = ab + ac for a, b, c ∈ F.
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(7)
Multiplication is commutative: ab = ba for each pair a, b in F.
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(8)
There exists an multiplicative identity denoted by 1 (not equal to 0) such that al = a for each a in F.
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(9)
For each a ∈ F, with a ≠ 0 there exists a multiplicative inverse denoted by a-1, such that aa-1 = 1.
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© 1997 Springer Science+Business Media New York
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Fine, B., Rosenberger, G. (1997). Complex Numbers. In: The Fundamental Theorem of Algebra. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1928-6_2
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DOI: https://doi.org/10.1007/978-1-4612-1928-6_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7343-1
Online ISBN: 978-1-4612-1928-6
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