Galois Theory - Tartarus

1 Part II Galois Theory Year201820172016201520142013201220112010 2009200820072006200547 Paper 4, Section II18I Galois TheoryLetKbe a field of characteristicp >0 and letLbe the splitting field of thepolynomialf(t) =tp t+aoverK, wherea K. Let Lbe a root off(t).IfL6=K, show thatf(t) is irreducible overK, thatL=K( ), and thatLis aGalois extension ofK. What is Gal(L/K)?Paper 3, Section II18I Galois TheoryLetLbe a finite field extension of a fieldK, and letGbe a finite group ofK-automorphisms ofL. Denote byLGthe field of elements ofLfixed by the action ofG.(a) Prove that the degree ofLoverLGis equal to the order of the groupG.(b) For any Lwritef(t, ) = g G(t g( )).(i) Suppose thatL=K( ). Prove that the coefficients off(t, ) generateLGoverK.(ii) Suppose thatL=K( 1, 2).

2 Prove that the coefficients off(t, 1) andf(t, 2) lie inLG. By considering the caseL=K(a1/21,a1/22) witha1anda2inK, or otherwise, show that they need not 2, Section II18I Galois TheoryLetKbe a field and letf(t) be a monic polynomial with coefficients inK. Whatis meant by asplitting fieldLforf(t) overK? Show that such a splitting field exists andis unique up to suppose thatKis a finite field. Prove thatLis a Galois extension ofKwithcyclic Galois group. Prove also that the degree ofLoverKis equal to the least commonmultiple of the degrees of the irreducible factors off(t) supposeKis the field with two elements, and letPn={f(t) K[t]|fhas degreenand is irreducible overK}.How many elements does the setP9have?Part II, 2018 List of Questions[TURN OVER201848 Paper 1, Section II18I Galois TheoryLetf(t) =t4+bt2+ct+dbe an irreducible quartic with rational briefly why it is that if the cubicg(t) =t3+ 2bt2+ (b2 4d)t c2hasS3as itsGalois group then the Galois group off(t) which prime numberspis the Galois group oft4+pt+pa proper subgroupofS4?]

3 [You may assume that the discriminant oft3+ t+ is 4 3 27 2.]Part II, 2018 List of Questions201848 Paper 2, Section II16I Galois Theory (a) Define what it means for a finite field extensionLof a fieldKto beseparable. ShowthatLis of the formK( ) for some L.(b) Letpandqbe distinct prime numbers. LetL=Q( p, q). ExpressLin theformQ( ) and find the minimal polynomial of overQ.(c) Give an example of a field extensionK6 Lof finite degree, whereLis not of theformK( ). Justify your 3, Section II16I Galois Theory (a) LetFbe a finite field of characteristicp. Show thatFis a finite Galois extensionof the fieldFpofpelements, and that the Galois group ofFoverFpis cyclic.(b) Find the Galois groups of the following polynomials:(i)t4+ 1 overF3.(ii)t3 t 2 overF5.(iii)t4 1 1, Section II17I Galois Theory (a) LetKbe a field and letf(t) K[t].

4 What does it mean for a field extensionLofKto be asplitting fieldforf(t) overK?Show that the splitting field forf(t) overKis unique up to isomorphism.(b) Find the Galois groups over the rationalsQfor the following polynomials:(i)t4+ 2t+ 2.(ii)t5 t II, 2017 List of Questions201749 Paper 4, Section II17I Galois Theory (a) State the Fundamental Theorem of Galois Theory .(b) What does it mean for an extensionLofQto becyclotomic? Show that a cyclotomicextensionLofQis a Galois extension and prove that its Galois group is Abelian.(c) What is the Galois groupGofQ( ) overQ, where is a primitive 7th root ofunity? Identify the intermediate subfieldsM, withQ6M6Q( ), in terms of ,and identify subgroups ofGto which they correspond. Justify your II, 2017 List of Questions[TURN OVER201744 Paper 2, Section II16H Galois Theory (a) LetK Lbe a finite separable field extension.]

5 Show that there exist onlyfinitely many intermediate fieldsK F L.(b) Define what is meant by anormalextension. IsQ Q(p1 + 7) a normalextension? Justify your answer.(c) Prove Artin s lemma, which states: ifK Lis a field extension,His a finitesubgroup of AutK(L), andF:=LHis the fixed field ofH, thenF Lis a Galoisextension with Gal(L/F) = 3, Section II16H Galois Theory (a) LetLbe the 13th cyclotomic extension ofQ, and let be a 13th primitive root ofunity. What is the minimal polynomial of overQ? What is the Galois group Gal(L/Q)?Put = +1 . Show thatQ Q( ) is a Galois extension and find Gal(Q( )/Q).(b) Define what is meant by aKummer extension. LetKbe a field of characteristiczero and letLbe thenth cyclotomic extension ofK. Show that there is a sequence ofKummer extensionsK=F1 F2 Frsuch thatLis contained 1, Section II17H Galois Theory (a) Prove that ifKis a field andf K[t], then there exists a splitting fieldLoffoverK.

6 [You do not need to show uniqueness ofL.](b) LetK1andK2be algebraically closed fields of the same characteristic. Showthat eitherK1is isomorphic to a subfield ofK2orK2is isomorphic to a subfield ofK1.[For subfieldsFiofK1and field homomorphisms i:Fi K2withi= 1,2, we say(F1, 1)6(F2, 2) ifF1is a subfield ofF2and 2|F1= 1. You may assume the existenceof a maximal pair (F, ) with respect to the partial order just defined.](c) Give an example of a finite field extensionK Lsuch that there exist , L\Kwhere is separable overKbut is not separable II, 2016 List of Questions201645 Paper 4, Section II17H Galois Theory (a) Letf=t5 9t+ 3 Q[t] and letLbe the splitting field offoverQ. Showthat Gal(L/Q) is isomorphic toS5. Let be a root off. Show thatQ Q( ) is neithera radical extension nor a solvable extension.

7 (b) Letf=t26+ 2 and letLbe the splitting field offoverQ. Is it true thatGal(L/Q) has an element of order 29? Justify your answer. Using reduction modptechniques, or otherwise, show that Gal(L/Q) has an element of order 3.[Standard results from the course may be used provided they are clearly stated.]Part II, 2016 List of Questions[TURN OVER201644 Paper 3, Section II16F Galois TheoryLetf Q[t] be of degreen>0, with no repeated roots, and letLbe a splittingfield forf.(i) Show thatfis irreducible if and only if for any , Rootf(L) there is Gal(L/Q) such that ( ) = .(ii) Explain how to define an injective homomorphism : Gal(L/Q) Sn. Find anexample in which the image of is the subgroup ofS3generated by (2 3). Find anotherexample in which is an isomorphism ontoS3.]

8 (iii) Letf(t) =t5 3 and assumefis irreducible. Find a chain of subgroups ofGal(L/Q) that shows it is a solvable group. [You may quote without proof any theoremsfrom the course, provided you state them clearly.]Paper 4, Section II17F Galois Theory (i) Prove that a finite solvable extensionK Lof fields of characteristic zero is aradical extension.(ii) Letx1,..,x7be variables,L=Q(x1,..,x7), andK=Q(e1,..,e7) whereeiare the elementary symmetric polynomials in the variablesxi. Is there an element Lsuch that 2 Kbut / K? Justify your answer.(iii) Find an example of a field extensionK Lof degree two such thatL6=K( )for any K. Give an example of a field which has no extension containing an 11thprimitive root of 2, Section II17F Galois Theory (i) State the fundamental theorem of Galois Theory , withoutproof.

9 LetLbe asplitting field oft3 2 Q[t]. Show thatQ Lis Galois and that Gal(L/Q) has asubgroup which is not normal.(ii) Let 8be the 8th cyclotomic polynomial and denote its image inF7[t] again by 8. Show that 8is not irreducible inF7[t].(iii) Letmandnbe coprime natural numbers, and let m= exp(2 i/m) and n= exp(2 i/n) wherei= 1. Show thatQ( m) Q( n) = II, 2015 List of Questions201545 Paper 1, Section II17F Galois Theory (i) LetK Lbe a field extension andf K[t] be irreducible of positive the theorem which states that there is a 1-1 correspondenceRootf(L) HomK(K[t]hfi,L).(ii) LetKbe a field andf K[t]. What is a splitting field forf? What does itmean to sayfis separable? Show that everyf K[t] is separable ifKis a finite field.(iii) The primitive element theorem states that ifK Lis a finite separable fieldextension, thenL=K( ) for some L.

10 Give the proof of this theorem II, 2015 List of Questions[TURN OVER201544 Paper 4, Section II18H Galois Theory (i) LetGbe a finite subgroup of the multiplicative group of a field. Show thatGiscyclic.(ii) Let n(X) be thenth cyclotomic polynomial. Letpbe a prime not dividingn,and letLbe a splitting field for noverFp. Show thatLhaspmelements, wheremisthe least positive integer such thatpm 1 (modn).(iii) Find the degrees of the irreducible factors ofX35 1 overF2, and the numberof factors of each 3, Section II18H Galois TheoryLetL/Kbe an algebraic extension of fields, andx L. What does it mean to saythatxis separable overK? What does it mean to say thatL/Kis separable?LetK=Fp(t) be the field of rational functions overFp.(i) Show that ifxis inseparable overKthenK(x) contains apth root oft.]