Abstract
In this chapter we prove that a finite field must have cardinality equal to a power of a prime. Such fields exist and we lay the grounds for the construction of such fields in Chap. 5. In this chapter we also prove a very important result that the multiplicative group of any finite field is cyclic. This makes it possible to define “discrete logarithms”-special functions on finite fields that are difficult to compute, and widely used in cryptography. We show that the Elgamal cryptosystem can also be based on the multiplicative group of a large finite field.
Oh field of battle, field of dying,
Who sank on you with glory here?
Ruslan and Liudmila. Alexander Pushkin (1799–1837)
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Notes
- 1.
See Sect. 3.1.3 for a brief historic note about Évariste Galois.
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Slinko, A. (2020). Fields. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-44074-9_4
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DOI: https://doi.org/10.1007/978-3-030-44074-9_4
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