What would the field corresponding to a Galois group of $S_5$ look like?

Question

I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since $S_5$ isn't solvable, constructing a field with a Galois group of $S_5$ with respect to $\mathbb{Q}$ can't be a radical extension, but is there some other function or operation that could be used? In other words, a quintic formula with radicals doesn't exist, but is there some function that isn't a purpose-built "this function yields solutions to a polynomial" function that could be used to solve quintics or higher polynomials?

Answer 1

I think this question may be a bit too vague as written, but here is a theorem one could appeal to in order to construct such a field extension. It is a fact that if $f \in \mathbb{Q}[x]$ is irreducible with $\deg f =p$ for some prime $p$ and has exactly 2 complex roots, then the Galois group of the splitting field extension is isomorphic to $S_{p}$.

Answer 2

Following up on the suggestion of Ariana. Let $$p(x)=x^5-3x^2+1,$$ when it is easy to show the Galois group of the splitting field of $p(x)$ over the rationals is $S_5$.

Assuming that after calculating the zeros of $p(x)=x^5-3x^2+1$ up to 200 decimal places makes it accurate enough, we can construct such a field $K$ as

$$K=\Bbb{Q}[x]/\langle f(x)\rangle,$$

where

$$f(x)=x^{120}+33750 x^{114}+1116000 x^{112}+3093750 x^{110}+1113560865 x^{108}+26921565000 x^{106}+604881000000 x^{104}+41167612108500 x^{102}+1431025760469805 x^{100}+14099233308750000 x^{98}+840755432509572240 x^{96}+29314339633598287500 x^{94}+838066825456291786500 x^{92}+17055343730004169787750 x^{90}+447099790210227436061835 x^{88}+16392596904188900682547500 x^{86}+524905496075736354421632150 x^{84}+7746834021558485383973580750 x^{82}+227400983306935610033504910210 x^{80}+5438203936818046114019096970000 x^{78}+161737233618167288338366207390005 x^{76}+3079140294664647133063741473941250 x^{74}+71589845771247618831377143880007030 x^{72}+1703874280827990134916438519652329750 x^{70}+37765960575062006944570272099577441065 x^{68}+671553629574346789742980518018455265750 x^{66}+13425663013791814923118775212295423224575 x^{64}+298316535549382310619276177470147839567500 x^{62}+6263322665754902028679045968192687021946417 x^{60}+113241137056665611046577975852478978853633750 x^{58}+2020482344097260796483156681326789886674641275 x^{56}+33607134753772181350504921906082658777563400000 x^{54}+550090827539075781453680855277159942042963975675 x^{52}+9645703546363285715097504428728612658988657633750 x^{50}+167589376219621416661551144351153197328368374710525 x^{48}+2591935957193938557000893428986050845007482057327500 x^{46}+37023543980552299462242594649218811608185860378407900 x^{44}+467204073589927694863531699904773814089803461948544000 x^{42}+5479652136885621906960743248094128630901237665341582235 x^{40}+60126566311045736532762730255787195693544124509832792500 x^{38}+622160800556257347660482211684147114157184154675733033935 x^{36}+5910280916004942423618805942747770827509420868385152115000 x^{34}+52106693211491045864419719017582977817447744056381867707300 x^{32}+432630947000379472605449339454013771610886258014994363460750 x^{30}+3410774365180782497074812580214046939106788562491870798046950 x^{28}+24817712899719418559089094578783591417923276718955280439927500 x^{26}+166698014114867874997691254686109786765856603463497453170639800 x^{24}+1049202472505126328549462956322404941429926499583868890532339750 x^{22}+6200097454408267204060687491470661741711907284917745068688392736 x^{20}+33063248055138491333939800416229050428492869440711652594093046250 x^{18}+160736778936639982917274226049744035868667050374165352429934665765 x^{16}+713664496954265807293435573649973706894373908524000311681369366250 x^{14}+2807053376943637105788469622562984789953566579091832894551656335585 x^{12}+9405202706949914238281707642961512160794793530858137713058729194750 x^{10}+29874334500240270890684107985135817331403471254102147813332994672740 x^8+70498858873024053142911958048672442662399336143969727445260372215750 x^6+152058512419218423143371634526878068936948757145171468466368924947025 x^4+244691421102204282324955998149483634966444221492246674851974602623750 x^2+191416245283609068525605381407966277298503971734124842355884163512481$$

has as its $120$ zeros the numbers $$x_\sigma:=2a_{\sigma(1)}+a_{\sigma(2)}-a_{\sigma(4)}-2a_{\sigma(5)}.$$ Here $a_1,\ldots,a_5$ are the zeros of the polynomial $p(x)$, and $\sigma$ ranges over all the permutations $\in S_5$.

The $120$ Galois conjugates of $x_{id}=2a_1+a_2-a_4-2a_5$ are all distinct. For otherwise there would be a linear dependency relation (over $\Bbb{Z}$) among the roots $a_i,i=1,2,3,4,5$, other than the Vieta relation $a_1+a_2+a_3+a_4+a_5=0$ — in violation of the fact that the splitting field has degree $120$.

Basic Galois correspondence then tells us that $\Bbb{Q}(x_{id})$ is the splitting field of $p(x)$. Hence all the numbers $x_\sigma$ are in there, and we can use any of them as a primitive element.