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Examination Problems Examination problems I conclude this book by a list of exercises and problems that students were really asked to solve for exams. Some of them are quite substantial. Exercise 7.1 proves a theorem of Selmer which shows that for any inte- ger n 2, the polynomial Xn − X − 1 is irreducible over Q. Exercise 7.2 is an elaboration on the casus irreductibilis: it explains and generalizes the fact that although the three roots of a polynomial equation of degree 3 might be real numbers, Cardan’s formulae use complex numbers. Exercise 7.5 is a theorem due to Galois about solvability by radicals of equations of prime degree. Exercise 7.11 proves a theorem due to E. Artin and O. Schreier about subfields F of an algebraically closed field Ω such that [Ω : F ] is finite. Exercise 7.1. Let n be any integer 2, and let S = Xn − X − 1. You shall show, following E. Selmer1,thatS is irreducible in Z[X]. a) Show that S has n distinct roots in C. b) For any polyomial P ∈ C[X]suchthatP (0) = 0, set m 1 ϕ(P )= z − , j z j=1 j where z1,...,zm are the complex roots of P , repeated according to their multiplicities. Compute ϕ(P ) in terms of the coefficients of P .Computeϕ(S). If P and Q are two polynomials in Q[X]withP (0)Q(0) =0,show that ϕ(PQ)=ϕ(P )+ϕ(Q). c) For any root z of S, establish the inequality 1 Math. Scand. 4 (1956), p. 287–302 182 Examination problems 1 1 2 z − > − 1. z |z|2 (Set z = reiθ and evaluate cos(θ)intermsofr.) m d) If x1,...,xm are positive real numbers with xj = 1, show that j=1 m xj m. j=1 e) Let P and Q be two polynomials in Z[X], of positive degrees, such that S = PQ.Showthat|P (0)| = 1 and that ϕ(P ) is a positive integer. Derive a contradiction, hence that S is an irreducible polynomial in Z[X]. Exercise 7.2. Let E be any subfield of the field R of real numbers. By def- inition, a real radical extension of E is a radical extension of E contained in R. a) Let E ⊂ F be any finite Galois extension, with F ⊂ R.Letα ∈ R such that αN ∈ E,whereN 2 is an integer, so that the extension E ⊂ E(α)is real radical, of exponent N. (i)Letm =[E(α):F ∩ E(α)] and set β = αm. Show that β belongs to F ∩ E(α). Deduce that F ∩ E(α)=E(β). (ii) Observe that some power of β belongs to E and show that [E(β):E] equals 1 or 2. b) Let E ⊂ F be a finite Galois extension, with F ⊂ R;letE ⊂ K be any real radical extension. Show that [K ∩ F : E] is a power of 2. (By induction: introduce L ⊂ K such that E ⊂ L is elementary radical, apply the induction hypothesis to the Galois extension L ⊂ FL and to the radical extension L ⊂ K.) c) Let P ∈ Q[X] be an irreducible polynomial of degree n, all the roots of which are real. Assume that one of the roots α of P belongstosomereal radical extension of Q. Show that n is a power of 2. √ √ Exercise 7.3. a) Show that the real numbers 1, 2and 5 are linearly independent over the field Q of rational numbers. √ √ b) We denote by K the field generated in R by 2and 5. What is the degree of the extension Q ⊂ K? c) Show that the extension Q ⊂ K is Galois and compute its Galois group. d) Find an element α ∈ K such that K = Q(α). e) Give the list of all subfields of K. Examination problems 183 Exercise 7.4. a) Let K be a field√ of characteristic zero, let d ∈ K such that d is not a square, and let K ⊂ K( d) be the quadratic extension generated by a square root of d in an algebraic closure of√K. Let x ∈ K. Show that x is a square in K( d) if and only if, either x is a square in K,ordx is a square in K. Through the end of this exercise, we shall consider three rational√ num- bers r, s and t. Assume that t is not a square in Q√. Denote by t one of the two complex√ square roots of t√and set E = Q( t). Assume moreover that r + s t is not a√ square in Q( t), denote by α one of the two complex square roots of r + s t,andsetF = E(α). b) Show that [F : E]=4.WhatistheminimalpolynomialoverQ of α? What are its conjugates in C? c) Show that the following are equivalent: ⊂ (i) the extension√ Q F is Galois; (ii) r − s t is a square in F ; (iii) there exists x ∈ Q such that r2 −s2t = x2 (first case) or r2 −s2t = tx2 (second case). d) Assume that the extension Q ⊂ F is Galois. In the first case, show that Gal(F/Q) (Z/2Z)2. In the second case, show that Gal(F/Q) Z/4Z. (Notice that an element of the Galois group is determined by the image of α, at least if s =0,andbemethodic.) √ e) (Numerical application√ )Write 5+ 21 without nested radicals. Is it possible for 7+2 5? Exercise 7.5. The aim of this exercise is to prove Proposition viii of Galois’s dissertation: “For an irreducible equation of prime degree to be solvable by rad- icals, it is necessary and sufficient that, any two of its roots being known, the others can be deduced rationally from them.” (French: “Pour qu’une ´equation irr´eductible de degr´e premier soit soluble par radicaux, il faut et il suffit que deux quelconques des racines ´etant connues, les autres s’en d´eduisent rationnellement.”) Part 1. Group theory a) Let X be a finite set, G a subgroup of the group S(X) of permutations of X.AssumethatG acts transitively on X.LetH be a normal subgroup in G. (i)Ifx and y are two elements in X, and if StabH (x) and StabH (y) denote −1 their stabilizers in H, show that there exists g ∈ G such that g StabH (y)g = StabH (x). Deduce that the orbits of x and y under the action of H have the same cardinality. 184 Examination problems (ii) We moreover assume that the cardinality of X is a prime number. If H = {1}, show that H acts transitively on X. b) Let p be a a prime number, and let Bp be the subgroup of permutations of Z/pZ that have the form m → am + b with a and b in Z/pZ. (i) What is the cardinality of Bp? (ii)ShowthatBp acts transitively on Z/pZ.Leth be any element in Bp which fixes two distinct elements of Z/pZ. Show that h =id. (iii)ShowthatBp is solvable. (You might want to introduce the subgroup of Bp consisting of permutations of the form m → m + b,withb ∈ Z/pZ.) c) Let p be a prime number and let G be a subgroup of the symmetric group Sp.AssumethatG is solvable and acts transitively on {1,...,p}.Let {1} = G0 G1 ··· Gm = G be a series of subgroups of G, where for each i, Gi is normal in Gi+1,with Gi+1/Gi a cyclic group of prime order. (i) Show that G1 is generated by some circular permutation of order p. −1 (ii) Show that there is σ ∈ Sp such that σ G1σ is generated by the circular permutation (1, 2,...,p). (iii)Identifytheset{1,...,p} with Z/pZ by associating to an integer its class modulo p. This identifies the group Sp of permutations of {1,...,p} with the group of permutations of Z/pZ. In particular, the group Bp of question b) is now viewed as a subgroup of Sp. −1 Show that σ Gσ is contained in Bp. Part 2. Field extensions Let E be a field, let P ∈ E[X] be an irreducible separable polynomial of prime degree p,andletE ⊂ F be a splitting extension of P . a) (i) Explain how the Galois group Gal(F/E) can be identified with a transitive subgroup of Sp. (ii) If Gal(F/E) is solvable, show that, α and β ∈ F being any two distinct roots of P ,thenF = E(α, β). b) (i) Show that Gal(F/E) contains a circular permutation of order p. (ii) Assume that there is a root α ∈ F of P such that F = E(α). Show that Gal(F/E) Z/pZ. Assume for the rest of this question that there are two distinct roots of P , α and β ∈ F ,suchthatF = E(α, β). (iii) Show that [F : E] p(p − 1). (iv) Show that Gal(F/E) contains a circular permutation of order p. (v)Letσ and τ be two circular permutations of order p in Gal(F/E). Show that they generate the same cyclic subgroup of order p, and conclude that this subgroup is normal in G. Examination problems 185 (vi) Show that Gal(F/E)issolvable. Exercise 7.6. a) Which of the following complex numbers are algebraic numbers?√ Give their√ minimal polynomial. 2, 1+ 2, (1 + π2)/(1 − π). √ b) Does there exist x ∈ Q( 3) whose square is 2? c) Let K ⊂ E be a quadratic extension ([E : K] = 2). Let P ∈ K[X]beany polynomial of degree 3.
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