Abstract
Let us recapitulate our results so far. We know that there is, for every prime p, a finite field with p elements, viz. the integers (mod p). This field we have denoted by F p . Furthermore, we know by Theorem 4.1 that provided there is an irreducible polynomial of degree m over F p , there exists a field with pm elements. On the other hand, by Theorem 5.1, there can be no finite field with a number of elements which is not a power of some prime number. This leaves two obvious questions:
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A.
For which values of p and m do irreducible polynomials exist?
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B.
Are there any other kinds of finite fields, i.e., ones that are not constructed by using Theorem 4.1?
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© 1987 Kluwer Academic Publishers
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McEliece, R.J. (1987). Finite Fields Exist and are Unique. In: Finite Fields for Computer Scientists and Engineers. The Kluwer International Series in Engineering and Computer Science, vol 23. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1983-2_6
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DOI: https://doi.org/10.1007/978-1-4613-1983-2_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-9185-5
Online ISBN: 978-1-4613-1983-2
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