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I'm currently studying the theory of Galois fields. And I have a question, what practical usage of this finite fields?

As stated in Wikipedia:

Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and Quantum error correction.

And what the other usage? I heard it used extensively in the image processing and recognition.

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Isn't that more or less also coding theory and error correction? –  Hagen von Eitzen Oct 19 '12 at 13:46
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That's not enough for you? Cryptography and coding theory are both eminently practical applications. –  Henning Makholm Oct 19 '12 at 13:47
    
Never though about image processing as a coding theory. Is it really have common principles? –  m0nhawk Oct 19 '12 at 13:50
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There are books on the matter in case you have access to a university library, for example "Applications of Finite Fields" and Lidl, R. and Niederreiter, H. Introduction to Finite Fields and Their Applications, rev. ed. Cambridge, England: Cambridge University Press, 1994. –  Amzoti Oct 19 '12 at 15:09
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All CD and DVD players use computations in Galois fields, as do many disk storage systems, applications that run on laptop computers, smart phones, tablets and the like. In other words, the usage is not only practical and Henning Makholm says, it is occurring all the time in your immediate vicinity. –  Dilip Sarwate Oct 19 '12 at 17:00
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up vote 5 down vote accepted

Finite fields are extensively used in design of experiments, an active research area in statistics that began around 1920 with the work of Ronald Fisher. Fisher was a major pioneer in the theory of statistics and one of the three major founders of population genetics.

I've heard of the use of finite fields in scheduling tournaments. Problems in that area may be the same mathematical problems as some of those that occur in design of experiments.

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