MATH8410Ex2.3Pete

# MATH8410Ex2.3Pete - SOLUTION TO MATH 8410 EXERCISE 2.4 PETE...

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SOLUTION TO MATH 8410, EXERCISE 2.4 PETE L. CLARK Prerequisites : This exercise uses Exercise 2.2 and Exercise 2.3. Statement : Let ( K, | | ) be a non-Archimedean normed ﬁeld, and let L/K be a purely transcendental extension. Show that | | extends to a norm on L . Background : We will review the deﬁnition of a purely transcendental extension. More signiﬁcantly, we will prove the result by transﬁnite induction , so let’s begin with a refresher 1 on that. Recall that a well-ordered set is a set S endowed with a total ordering rela- tion (a reﬂexive, anti-symmetric, transitive relation such that for all x, y S , at least one of x y and y x holds) which has the additional property that every nonempty subset T of S has a least element. For instance, the natural numbers N with their usual ordering are a well-ordered set, as are N ∪ {∞} , the natural numbers with an additional element such that x < for all x N . In fact, for any well-ordered set S , it turns out to be useful to deﬁne the well-ordered set S + = S ∪{∞} in the same way. (In particular, S + has a maximal element, whereas S need not.) The empty set is well-ordered. Any nonempty well-ordered set has a least ele- ment, which we might as well call 0. 2 If S is well-ordered, and x is any element of S except possibly the maximal element (if any), then the set { y S | x < y } is nonempty hence has a least element, the successor of x . In the natural numbers, the successor of n is nothing else than n +1, so we may as well denote the successor of x by x + 1 in the general case. An element

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MATH8410Ex2.3Pete - SOLUTION TO MATH 8410 EXERCISE 2.4 PETE...

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