F×

F× denotes the multiplicative group of a field F. It consists of all nonzero elements of F equipped with multiplication. Because multiplication in a field is commutative, F× is an abelian group; its identity element is 1 and every nonzero element a has an inverse a−1.

If F is finite with q elements, then F× has q−1 elements and is cyclic. In other words, there exists an element g in F× such that every nonzero element of F is a power of g. This property is central to finite field theory and underpins many algorithms in coding theory and cryptography.

For the real numbers R, the multiplicative group R× consists of the nonzero real numbers under multiplication. It has two connected components, corresponding to positive and negative numbers. The subgroup of positive reals is isomorphic to (R, +) via the logarithm map, so R× ≈ {±1} × (0, ∞) and -1 has order 2.

For the complex numbers C, C× is infinite and not cyclic. Each nonzero complex number z can be written uniquely as z = r e^{iθ} with r > 0 and θ real, giving a standard decomposition C× ≈ (0, ∞) × S^1, where S^1 is the unit circle. The finite-order elements of C× are the roots of unity on S^1.

Summary: F× is the fundamental multiplicative structure of a field, capturing all nonzero elements under multiplication. Its structure varies by F: finite fields yield cyclic groups of order q−1, while the real and complex fields yield infinite, noncyclic groups with characteristic decompositions into magnitude and phase.