Real-world applications of fields, rings and groups in linear algebra.

Well, I'd consider the brand new book from Gilbert Strang: ''Linear Algebra and Learning from Data'', Cambridge Univ. Press, 2018. His learning from data culminates into the construction of deep neural networks.

Another application besides cryptography is coding theory, where finite fields are used to define linear codes. There is also a decent generalization to linear codes over the ring of integers modulo 4, which give rises by the Gray map to nonlinear binary codes which are better than any linear code with the same block length.