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Diffusion cloud chamber with tracks of ionizing radiation (alpha particles) that are made visible as strings of droplets. In dosimetry, linear energy transfer (LET) is the amount of energy that an ionizing particle transfers to the material traversed per unit distance.
Let / be a field extension.An element is a primitive element for / if = (), i.e. if every element of can be written as a rational function in with coefficients in .If there exists such a primitive element, then / is referred to as a simple extension.
This shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.
The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
If a 1 = 0, then R 3 (y) = y 3 + 2a 2 y 2 + (a 2 2 − 4a 0)y. The roots of this polynomial are 0 and the roots of the quadratic polynomial y 2 + 2 a 2 y + a 2 2 − 4 a 0 . If a 2 2 − 4 a 0 < 0 , then the product of the two roots of this polynomial is smaller than 0 and therefore it has a root greater than 0 (which happens to be − a 2 + 2 ...
The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let J α be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in J α
If x is an algebraic number then a n x is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where a n x n is the highest-degree term of p(x). The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients.
For polynomials with real coefficients, it is often useful to bound only the real roots. It suffices to bound the positive roots, as the negative roots of p(x) are the positive roots of p(–x). Clearly, every bound of all roots applies also for real roots. But in some contexts, tighter bounds of real roots are useful.