By B.H. Gross, B. Mazur

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CHAPTER I 21 The element X + V* lies in the left kernel if XY = 0 for all Y £ V. This means X£ V*, and hence X+ V* = V*, the zero element of V /V*. The left kernel is now zero. The right kernel is obviously the old W*. 9) tells us that the codimension of W* in V is the same as the dimension of our left factor which is W /V*. 10) dim V/W* =dim W/V*. Suppose now that both kernels V* and W* are zero and that dim W is finite. 10) shows that dim V is also finite and equals dim W. In this case we can use all our results unrestrictedly on subspaces of both Wand V.

Then we know that k is the disjoint union -PV {0} VP and that P +PCP. These are the first two axioms of an ordered field. 15. Suppose a, b e P. Then a is on the same side of 0 as 1. Multiplying by b we get that ab is on the same side of 0 as b. But this means ab t P. The field k is ordered. We see that the maps x--+ ax (for a ¢ O) also either preserve or reverse the ordering. 16. If a field k is weakly ordered and has more than two elements, then it i8 an ordered field. Hilbert has constructed an ordered non-commutative field and we shall present his example.

33) a"d" = d"a" and c"b" = b"c". 34) ((a + c)(b + d))" = + c)"(b + d)" = a"b" ) ~(b + a"d" + c"b" + c"d" or + d)"(a +c)" = b"a" + d"a" + b"c" + d"c". A direct computation of the left side gives: (ab)" + (ad)" + (cb)" + (cd)" = a"b" + a"d" + c"b" + d"c". 27). 27) again. 26) we either have the first possibility for all a, b E k or else the second one for all a, b E k. 9. 14. A field k is said to be ordered, if, first of all, it is ordered as an additive group. 11 is not necessary since addition is commutative) this means that a set P of so-called "positive" elements is singled out such that 1) k = -P U {0} UP (disjoint), 2) P + P C P (Pis closed under addition).