irreducible polynomial sentence in Hindi
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- It is helpful to compare irreducible polynomials to prime numbers : prime numbers ( together with the corresponding negative numbers of equal magnitude ) are the irreducible integers.
- Since the substitution is an automorphism of the ring, the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally.
- Since the substitution is an automorphism of the ring, the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally.
- One of these first results was irreducible polynomial " f " with integer coefficients such that " f " ( ? ) = 0.
- If denotes the finite field of order ( where is necessarily a prime power ), then the number of monic irreducible polynomials of degree over is given by:
- To ensure that this field is actually-dimensional and does not collapse to an even smaller field, it is sufficient that is an irreducible polynomial over the rationals.
- Thus for any irreducible polynomial whose degree is not a power of 2 and which has all roots real, the roots cannot be expressed purely in terms of real radicals.
- They exhibit many of the general properties of the concept of " irreducibility " that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors.
- This implies that " p " is an irreducible polynomial, and thus that the quotient ring K [ X ] / \ langle p \ rangle is a field.
- For p = 2 and p = 3, it's not difficult to get a desired factorization into irreducible polynomials ( which can be prime, by the hypothesis ).
- Let p ( x ) be an irreducible polynomial of degree m over \ mathbb { F } _ 2, and let p be the period of p ( x ).
- Irreducible polynomials over finite fields are also useful for Pseudorandom number generators using feedback shift registers and discrete logarithm over "'F "'2 " n ".
- When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials ( see Factorization of polynomials ).
- This means that every element of these rings is a product of a constant and a product of irreducible polynomials ( those that are not the product of two non-constant polynomials ).
- In fact, if is a prime number, and, the field of order is a simple algebraic extension of the prime field of elements, generated by a root of an irreducible polynomial of degree.
- Every irreducible closed subset of "'P " "'n " of codimension one is a hypersurface; i . e ., the zero set of some homogeneous irreducible polynomial.
- Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing " P ", but leaving " L " the same.
- It follows that the th cyclotomic polynomial factors over into distinct irreducible polynomials that have all the same degree, say, and that is the smallest field of characteristic that contains the th primitive roots of unity.
- The minimal polynomial of an algebraic element " x " of " L " is irreducible, and is the unique monic irreducible polynomial of which " x " is a root.
- For example, let " P " be an irreducible polynomial with integer coefficients and " p " be a prime number which does not divide the leading coefficient of " P ".
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