Intuition
Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and p-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution?
One can ask this for other rings or fields: integers, for instance, or number fields. For number fields, rather than reals and p-adics, one uses complex embeddings and -adics, for prime ideals .
Read more about this topic: Hasse Principle
Famous quotes containing the word intuition:
“All appeared new, and strange at first, inexpressibly rare and delightful and beautiful. I was a little stranger, which at my entrance into the world was saluted and surrounded with innumerable joys. My knowledge was divine. I knew by intuition those things which since my Apostasy, I collected again by the highest reason.”
—Thomas Traherne (1636–1674)
“Reason is sight. Instinct is touch. Intuition is smell.”
—Mason Cooley (b. 1927)
“Well, intuition isn’t much help in police work. Facts are what we need.”
—Crane Wilbur (1889–1973)