So it will not be true that " any element in GF ( p ) [ x ] / f ( x ) other than the additive identity " will generate the multiplicative group.
32.
This can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraic torus ( equivalently by a multiplicative group of torus embeddings ".
33.
The multiplicative group of " p "-adic units is a product of the finite group of roots of unity and a group isomorphic to the " p "-adic integers.
34.
It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in " k " form a subgroup of finite index in the multiplicative group.
35.
But a finite group ( the multiplicative group of " D " in our case ) cannot be a union of conjugates of a proper subgroup; hence, " n " = 1.
36.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers ( whose Lie algebra is the additive group of all real numbers ).
37.
Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number g, resulting in gS being the interval with bounds g \ cdot a, g \ cdot b.
38.
Here " G " m ( the multiplicative group ) denotes the sheaf of invertible regular functions, and the shift [ " 1 ] means that this sheaf is viewed as a complex in degree 1.
39.
There are two kinds of spheres in the theory : those coming from the multiplicative group playing the role of the-sphere in topology, and those coming from the simplicial sphere ( considered as constant simplicial sheaf ).
40.
Formally, " g " is a group ( typically the multiplicative group of a finite field or an elliptic curve group ) and " x " and " y " are randomly chosen integers.
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