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A detailed definition is given here. Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state. In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball.
Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's Last Theorem. If a polynomial g ( x ) ∈ Z [ x ] {\displaystyle g(x)\in \mathbb {Z} [x]} is a perfect square for all large integer values of x , then g(x) is the square of a polynomial in Z [ x ] . {\displaystyle \mathbb {Z} [x].}
Equivalently, for this definition, an irreducible polynomial is an irreducible element in a ring of polynomials over R. If R is a field, the two definitions of irreducibility are equivalent. For the second definition, a polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that both have a ...
Because of this problem of undecidability in the formal language of computation, Wolfram terms this inability to "shortcut" a system (or "program"), or otherwise describe its behavior in a simple way, "computational irreducibility." The idea demonstrates that there are occurrences where theory's predictions are effectively not possible.
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for this property.
The Riemann Hypothesis. Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize ...
Consider the polynomial Q(x) = 3x 4 + 15x 2 + 10.In order for Eisenstein's criterion to apply for a prime number p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10.
Absolute irreducibility more generally holds over any field not of characteristic two. In characteristic two, the equation is equivalent to (x + y −1) 2 = 0. Hence it defines the double line x + y =1, which is a non-reduced scheme. The algebraic variety given by the equation + = is not absolutely irreducible.