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Gps Abh1.Pdf 1 Lecture L6: Pause and Refresh By reading the following summaries of the first five lectures, the rest of the book may become intelligible without studying the details of those lectures. In the first lecture we have introduced the basic structures of algebra such as groups, rings, fields, vector spaces, ideals, modules, polynomials, rational functions, euclidean domains, principal ideal domains, and unique factorization domains. In the second lecture, after introducing power series, meromorphic series, and valuations, we show the equivalence of well-ordering and Zorn's Lemma and use them to establish the existence of vector space basis, transcendence basis, algebraic closure, and maximal ideals. The third lecture deals with the power series theorems of Newton, Hensel, and Weierstrass. The fourth and fifth lectures deal with ideals, modules, varieties, and models which are the avatars of varieties full of local rings. x1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS For sets (= collections of objects) S and T , a map φ : S ! T is an assignment which to every x 2 S, i.e., to every element (= object) x of S, assigns φ(x) 2 T ; this may be written x 7! φ(x); the element φ(x) is called the image of x under φ; we put dom(φ) = S and ran(φ) = T and call these the domain and range of φ respectively. The composition of maps φ : S ! T and : T ! U is the map φ : S ! U given by ( φ)(x) = (φ(x)). The map φ is injective, or is an injection, if φ(x) = φ(y) ) x = y.A subset of S is a set R whose objects are amongst the objects of S; we write R ⊂ S; we may also write S ⊃ R and call S an overset of R. We put φ(R) = fφ(x): x 2 Rg = the set of all φ(x) with x varying over R, and call this the image of R under φ. We put im(φ) = φ(S) and call this the image of φ; the map φ is surjective, or is a surjection, if φ(S) = T . The map φ is bijective, or is a bijection, if it is injective as well as surjective. For a bijection φ : S ! T we have the inverse map φ−1 : T ! S given by φ−1(y) = x , φ(x) = y. Without assuming the map φ : S ! T to be bijective, for any y 2 T we put φ−1(y) = fx 2 S : φ(x) = yg, and for any U ⊂ T we put φ−1(U) = fx 2 S : φ(x) 2 Ug; moreover, for any A ⊂ S and B ⊂ T with φ(A) ⊂ B, by φj(A;B) we denote the map A ! B which sends every x 2 A to φ(x) in B, and we call this the restriction of φ to (A; B); if B = φ(A) then we may denote this by φjA and call it the restriction of φ to A. The empty set is denoted by ;. For subsets R1 and R2 of a set S, the com- plement of R2 in R1 is denoted by R1 n R2, i.e., R1 n R2 = fx 2 R1 : x 62 R2g; needless to say that x 62 R2 means x is not an element of R2, just as x 6= y means x is not equal to y, and so on. For subsets R1 and R2 of a set S, their intersection is denoted by R1 \ R2 and their union is denoted by R1 [ R2, i.e., R1 \R2 = fx 2 S : x 2 R1 and x 2 R2g and R1 [R2 = fx 2 S : x 2 R1 or x 2 R2g. Similarly for more than two subsets R1;:::;Rm of a set S we have \1≤i≤mRi = fx 2 R : x 2 Ri for all ig and [1≤i≤mRi = fx 2 S : x 2 Ri for some ig. More generally we could have a family (Ri)i2I of subsets Ri of a set S indexed by an indexing set I, i.e., i 7! Ri gives a map of I into the set of all subsets of S, and then we put \i2I Ri = fx 2 S : x 2 Ri for all i 2 Ig and [i2I Ri = fx 2 S : x 2 Ri for some i 2 Ig.A partition of a set S is a collection of nonempty subsets of S such that S is their union and any two of them have an empty intersection. 2 x1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS 3 The set whose elements are x1; : : : ; xe (which may or may not be distinct) is denoted by fx1; : : : ; xeg. By N+ ⊂ N ⊂ Z ⊂ Q ⊂ R ⊂ C we denote the sets of all positive integers, nonnegative integers, integers, rational numbers, real numbers, and complex numbers respectively. The size of any set S, i.e., the number of elements in it, is denoted by jSj; note that then jSj 2 N or jSj = 1 according as S is finite or infinite; later on we shall give a more precise meaning to different types of infinities, and then jSj will denote the \cardinal number" of S. Clearly Q j;j = 0. For Sn defined below we have jSnj = n! = 1≤i≤n i, and hence jS0j = 1; convention: an empty product is 1 and an empty sum is 0. A group G is a set with a binary operation which to every pair of elements x; y in G associates a product xy 2 G such that: (i) (xy)z = x(yz) for all x; y; z in G (associativity); (ii) there is 1 2 G with 1x = x1 = x for all x 2 G (existence of identity); (iii) for every x 2 G there is x−1 2 G with xx−1 = x−1x = 1 (existence of inverse). A subgroup of a group G is a subset H which is a group under the same operation as G; we then write H ≤ G. If H ≤ G with H 6= G then we write H < G and call H a proper subgroup of G. A normal subgroup of a group G is a subgroup H such that for all x 2 G we have xHx−1 = H where xHx−1 = fxyx−1 : y 2 Hg; we then write H/G. If H/G with H 6= G, then H is called a proper normal subgroup of G. A group G is simple if G 6= 1 and G has no proper normal subgroup 6= 1, where 1 denotes the identity group having only one element. The size jGj of a group is its order; the order of x 2 G is the smallest r 2 N+ with r x = 1; if there is no such r 2 N+ then the order of x is 1. A group G is cyclic if it is generated by a single element x, i.e., if every element of G is a power of x; we then denote G by Zr where r is the order of x which may be 1; clearly Zr is simple , r is a prime number. A homomorphism of a group G into a group J is a map φ : G ! J such that φ(1) = 1 and φ(xy) = φ(x)φ(y) for all x; y in G; the kernel of φ is defined by ker(φ) = φ−1(1), and we have im(φ) ≤ J and ker(φ) /G; note that φ is injective iff (= if and only if) ker(φ) = 1 and when that is so we call φ a monomorphism; if φ is surjective, i.e., if im(φ) = J, then we call φ an epimorphism; if φ is bijective then we call it an isomorphism; if J = G and φ is an isomorphism then we call φ an automorphism of G. For H ≤ G we put xH = fxy : y 2 Hg and call this a left coset of H in G (similar definition of a right coset), and by G=H we denote the set of all left cosets of H in G, and note that this is a partition of G; also we put [G : H] = jG=Hj and call this the index of H in G. If H/G then G=H becomes a group by defining (xH)(yH) = (xy)H, and we call G=H the factor group of G by H; now x 7! xH gives an epimorphism G ! G=H with kernel H; we call this the canonical epimorphism of G onto G=H. A finite group G is solvable if there is a chain 1 = G0 /G1 / ··· /Gr = G such that Gi=Gi−1 is cyclic of prime order for 1 ≤ i ≤ r. The set of all bijections of a set S onto itself forms a group under composition which we call the symmetric group on S and denote it by Sym(S). If jSj = n 2 N then Sym(S) = Sn. Any permutation σ on n letters, i.e., σ 2 Sn, can be written as a product of a certain number ν of transpositions, and the parity of ν, i.e., its evenness or oddness, depends only on σ; we call the permutation σ even or odd 4 LECTURE L6: PAUSE AND REFRESH according as ν is even or odd, and we define the signature sgn(σ) of σ to be 1 or −1 according as σ is even or odd; note that a transposition is an element τ in Sn = Sym(S) such that for some i 6= j in S we have τ(i) = j, τ(j) = i, and τ(l) = l for all l 2 S n fi; jg.
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