We now establish some further results, of general theoretical interest.

Jurisdiction concerns the power of the state under international law to regulate or otherwise impact upon people, property and circumstances and reflects the basic principles of state sovereignty, equality of states and non-interference in domestic affairs. Jurisdiction is a central feature of state sovereignty, for it is an exercise of authority which may alter or create or terminate legal relationships and obligations. It may be achieved by means of legislative, executive or judicial action. In each case, the recognised authorities of the state as determined by the legal system of that state perform certain functions permitted them which affect the life around them in various ways. In the United Kingdom, Parliament passes binding statutes, the courts make binding decisions and the administrative machinery of government has the power and jurisdiction (or legal authority) to enforce the rules of law. It is particularly necessary to distinguish between the capacity to make law, whether by legislative or executive or judicial action (prescriptive jurisdiction or the jurisdiction to prescribe), and the capacity to ensure compliance with such law, whether by executive action or through the courts (enforcement jurisdiction or the jurisdiction to enforce). Jurisdiction, although primarily territorial, may be based on other grounds, for example nationality, while enforcement is restricted by territorial factors.

The evolution of the modern nation-state and the consequent development of an international order founded upon a growing number of independent and sovereign territorial units inevitably gave rise to questions of international cooperation. The first major instance of organised international cooperation occurred with the Peace of Westphalia in 1648, which ended the thirty-year religious conflict of Central Europe and formally established the modern secular nation-state arrangement of European politics. Over a century later, the Napoleonic wars terminated with the Congress of Vienna in 1815, marking the first systematic attempt to regulate international affairs by means of regular international conferences. The Congress system lasted, in various guises, for practically a century and institutionalised not only the balance of power approach to politics, but also a semi-formal international order.

Suppose that 𝐾 is a field whose characteristic is not 2, that 𝑓 is a normal separable monic polynomial of degree 𝑛 in 𝐾[𝑥] and that 𝐿 : 𝐾 is a splitting field extension for 𝑓. Then the Galois group Γ[𝐿 : 𝐾] acts on the set {α1, …, α𝑛} of roots of 𝑓 in 𝐿, and can therefore be identified with a subgroup of Σ𝑛.

The second main topic of Galois theory is the study of polynomials. The collection of all polynomials with integral coefficients forms an integral domain, and integral domains provide an appropriate setting for the study of divisibility and factorization.