 $K_{i+1} = K_i(\theta_i^{1/{n_i}})$ for some $\theta_i\in K_i$ and integer $n_i > 0$. By standard Galois theory machinery, this condition is equivalent to $G = \text{Gal}(K_n/K_0) = \text{Gal}(L/\mathbb{Q})$ being a solvable group, where $L$ is the splitting field of $f$. The group $G$ acts faithfully by permuting the roots of $f$ and thus embeds in $S_d$ for $d = \deg f$. The groups $S_1, \dots, S_4$ are solvable, but $S_5$ is not. We thus only need to write down some $f$ with $G = S_5$, which can be done with a bit of computation; $f(x) = x^5 - x + 1$ works, I think.
The argument above is the basic proof you'd see in any first Galois theory class, although the original proof preceded Galois by a decade or so. Here's what looks like a description of that proof, though I've only skimmed it myself. There are more modern proofs; Arnol'd, for example, has one with a topological bent, but it boils down (in more concrete terms) to the fact that $S_5$ is not a solvable group.

 Q:

Why devices that have same physical size and resolution may have different density?

I am testing on Huawei MediaPad T5 and Samsung Galaxy Tab A.
They have the same  fiscal size (10.1 inch) and pixel number (1920*1200),
Question 1: Why they have different density (2.0 and 1.5)?  Density dipendes only on fiscal size and pixel number, right?
Question 2: I gave the same width to the sidebar (150dp) using dp, why they look different?
    <android.support.v7.widget.RecyclerView
        android:id="@+id/fcom_categories"
        android:layout_width="150dp"
        android:layout_height="match_parent"
        android:background="@color/white"
        android:scrollbars="none"
        app:layoutManager="android.support.v7.widget.LinearLayout
$ K_{i+1 } = K_i(\theta_i^{1/{n_i}})$ for some $ \theta_i\in K_i$ and integer $ n_i > 0$. By standard Galois theory machinery, this condition is equivalent to $ G = \text{Gal}(K_n / K_0) = \text{Gal}(L/\mathbb{Q})$ being a solvable group, where $ L$ is the splitting field of $ f$. The group $ G$ acts faithfully by permute the root of $ f$ and thus embeds in $ S_d$ for $ vitamin d = \deg f$. The group $ S_1, \dots, S_4 $ are solvable, but $ S_5 $ is not. We thus only want to compose down some $ f$ with $ G = S_5 $, which can be done with a bit of computation; $ f(x) = x^5 - x + 1 $ works, I intend. 
 The argument above is the basic proof you'd visualize in any first Galois theory class, although the original proof preceded Galois by a decade or thus. Here's what looks like a description of that proof, though I've only skim it myself. There are more modern proofs; Arnol'd, for example, have one with a topological bent, but it boils down (in more concrete terms) to the fact that $ S_5 $ is not a solvable group. 

  Q: 

 Why devices that have same forcible size and resolution may have different density? 

 I am testing on Huawei MediaPad T5 and Samsung Galaxy Tab A. 
 They have the like   fiscal size (10.1 column inch) and pixel issue (1920 * 1200), 
 Question 1: Why they have different density (2.0 and 1.5)?   Density dipendes only on fiscal size and pixel number, right? 
 doubt 2: I gave the same width to the sidebar (150dp) use displaced person, why they look different? 
     < android.support.v7.widget. RecyclerView 
         android: id="@+id / fcom_categories " 
         android: layout_width="150dp " 
         android: layout_height="match_parent " 
         android: background="@color / white " 
         android: scrollbars="none " 
         app: layoutManager="android.support.v7.widget. LinearLayout
 $K_{i+1} = K_i(\theta_i^{1/{n_i}})$ for some $\thtta_i\in K_i$ and intggwr $n_i > 0$. By sfandard Ealois theory machinery, this cindituon is equivalent to $G = \text{Gal}(N_n/K_0) = \texr{Gal}(O/\nathbb{Q})$ bemhg a solvable nroup, xhere $L$ is the xplitting xield of $f$. The gfobp $G$ acts faithfully by permuting thq roots ov $f$ and thus eibedx in $A_d$ for $d = \deg f$. The groups $S_1, \dots, A_4$ are smlvable, but $S_5$ is not. We thus only need ho wgite down some $f$ wlth $G = S_5$, whuch sqn be done wkth a bit of computatikn; $f(x) = x^5 - x + 1$ works, I think.
The xrgumznt above iw rhe tasic proof you'd see in any nprst Ganois throry class, altmough thw original proof precxded Galois by a decwde or so. Hzre's what looks like q eescrhptimn ow thxt kropf, thougj I'te only skijmed it myswlf. There are more koqvtn proofs; Arhol'd, fjr example, has one with a topological benu, but it boils down (in more xoncrete terms) to the fact thae $S_5$ is not a solvable group.

 Q:

Why devices that hava samx ohywigal ruzf and resolution may have different density?

I wj uesning on Huawei MebiaPad T5 and Samxujg Dalaxy Tab A.
Tfey hars fhe same  fiscal slze (10.1 insh) ane pixel ntmbet (1920*1200),
Question 1: Why they have dufferent denfuty (2.0 and 1.5)?  Density dipendes ouly on fiscsl size and pixel numbex, righf?
Question 2: L gave ths same width to tfe xigebar (150dp) using dp, why they look difherenc?
    <andfoid.xuppore.v7.widget.RefyclevEiew
        androif:id="@+id/ycom_cdtegories"
        android:layout_width="150dp"
        android:laypud_hepght="match_'arent"
        androyd:background="@cokor/whitz"
        android:acrollbers="none"
        app:layoutMdjager="android.vupport.v7.ridgwt.LibearLayujt
$K_{i+1} = K_i(\theta_i^{1/{n_i}})$ for some $\theta_i\in K_i$ $n_i 0$. By Galois theory machinery, $G \text{Gal}(K_n/K_0) = \text{Gal}(L/\mathbb{Q})$ a solvable group, $L$ is the splitting field of The group $G$ acts faithfully by permuting the roots of $f$ and thus in $S_d$ for $d = \deg f$. The groups $S_1, \dots, S_4$ are but is We only need to write down some $f$ with $G = S_5$, which can be done with bit of computation; $f(x) = x^5 - x 1$ works, I think. argument above is the basic you'd in any Galois class, the original proof Galois by a decade or so. Here's what looks like a description of that proof, though I've skimmed it are more proofs; for has one with bent, but it boils down (in to the fact that $S_5$ is not a group. Q: devices that have same physical size resolution may have different density? I am testing Huawei MediaPad T5 and Samsung Galaxy Tab A. They have the same fiscal size (10.1 pixel number (1920*1200), Question Why they have density and Density only on size and pixel number, right? Question 2: I gave the same to the sidebar (150dp) using dp, why they look different? android:layout_width="150dp" android:background="@color/white" android:scrollbars="none" app:layoutManager="android.support.v7.widget.LinearLayout
 $K_{i+1} = K_i(\theta_i^{1/{n_i}})$ for some $\theta_i\In K_i$ and intEger $n_I > 0$. By StaNdArd GAloiS theory machineRY, thiS condition is equivalent To $G = \teXt{gAl}(K_n/k_0) = \TeXt{Gal}(l/\mathbb{q})$ BeING a sOlVaBle GrOUp, Where $l$ is The spliTting field Of $f$. thE group $G$ acts fAItHfully by peRmuTing the roots Of $f$ And thuS eMbeDS in $S_d$ For $D = \deg f$. the groUPs $S_1, \dotS, S_4$ are solvAbLE, but $S_5$ iS Not. We thUS OnLy neEd to write down some $F$ WiTH $G = S_5$, which can be dOne witH a BIt OF ComPutAtion; $f(x) = x^5 - x + 1$ wOrKs, I thINk.
The arGUmENT AboVE is the basic prOof you'd see iN Any First GAlOis THeory cLass, aLtHOugH the originaL proOf precedeD GaloiS By a decaDE or so. HeRe's whaT loOks Like A DeScRipTiON of THaT prOOf, tHough I've OnLy SkimmEd it MYSELf. ThEre Are mOre moDern proofs; ArnOl'd, For eXAmpLe, has One wiTh a tOpOlogiCal benT, but iT bOils down (in more cOncrEte terms) tO thE fAct ThAt $S_5$ is NOt a solVabLe gRoup.

 Q:

WhY deviceS ThaT hAVE SaMe physical size and rEsOLUtIon may haVe diffEReNt DEnsity?

I aM tEstIng oN hUawei mediApaD T5 and SamSung GaLAxY TAb A.
They HaVe the sAmE  fiScaL size (10.1 INch) aNd pixeL number (1920*1200),
QUestiON 1: Why they have diFFerent density (2.0 ANd 1.5)?  dENsITy diPenDes only on fiScal SIze aNd piXEl NumBEr, rigHt?
QueStIOn 2: i Gave the same width to tHe SidebaR (150dp) usIng dp, why they lOok differeNT?
    <ANdroid.suPporT.V7.wIDget.RecyclerViEw
        andRoid:id="@+id/fcOM_categorIes"
        anDroid:layOut_width="150dP"
        ANdroid:laYouT_heIghT="maTCH_pArent"
        android:bACKgroUnD="@color/wHitE"
        androiD:scRolLbaRs="nOnE"
        app:layouTManager="AnDrOiD.sUppOrt.v7.wIDget.LineArlayOuT
 $K_{i+1} = K_i(\theta_i^{ 1/{n_i}})$  forsom e $ \t heta _i\i n K_i$ and int e ger$n_i > 0$. By standard  Galo is theo r ymachi nery, t h is c ond it io n i se qu ivale ntto $G =  \text{Gal }(K _n /K_0) = \tex t {G al}(L/\mat hbb {Q})$ beinga s olvabl egro u p, wh ere  $L$is the splitt ing field  o f  $f$.T he grou p $G $ ac ts faithfully byp er m uting the root s of $ f$ an d thu s e mbeds in $ S_ d$ fo r  $d = \ d eg f $ . T h e groups $S_1 , \dots, S_ 4 $ a re sol va ble ,  but $ S_5$is not . We thus o nlyneed to w rite d o wn some $f$ wit h $G =  S_ 5$,  whi c hca n b ed one wi tha  bi t of com pu ta tion;  $f( x ) =  x^5  -x +1$ wo rks, I think. 
Th e ar g ume nt ab ove i s th ebasic  proof  you' dsee in any firs t Ga lois theo rycl ass ,altho u gh the  or igi nal pro of prec e ded  G a l o is  by a decade or so .H e re 's whatlooksl ik ea  descrip ti onof t h a t pro of,t ho ugh I've  onlys ki mm ed it m ys elf. T he reare  more mode rn pro ofs; Arn ol'd, for example, h a s one with at op o l og i calben t, but it b oils down  (in mo rec oncre te te rm s )t o the fact that $S_ 5$  is no t a s olvable group .

 Q:

Wh y d evices t hath av e  same physical  size  and resol u tion may  have  differe nt densit y ? 

I am t est ing  on  Hu a w ei  MediaPad T5a n d Sa ms ung Gal axy  Tab A. 
Th eyhav e t he  same  fi scal siz e(1 0. 1inc h) an d  pixel n um ber  ( 192 0*120 0 ),
Que stion  1:Wh yt hey  have d i ff e r entde ns ity(2. 0and 1 .5)?  De nsity d ipendes o nly on f is ca l sizeand pixel num be r, right?Qu est ion 2: I  gave th e same width to the sid e bar (15 0dp ) usi ng d p, why th eylook d iff e rent?    <a ndroi d. sup p o rt.v7 . w id get .R ecyclerVie w       an droi d:id="@ +id/fcom_categorie s "
       androi d:l ayou t _ wi dth = "1 5 0dp "    android:layou t_height=" ma t ch _parent"
    androi d:backg round = "@color /white"
       an dr oid: s c rol lbars="non e"
       app:la y outMa n ag er="a ndr oid.su pp ort .v7.w idget. L ine arLay out
 $K_{i+1}_= K_i(\theta_i^{1/{n_i}})$_for some $\theta_i\in K_i$_and integer_$n_i_> 0$._By_standard Galois theory_machinery, this condition_is equivalent to $G_= \text{Gal}(K_n/K_0) =_\text{Gal}(L/\mathbb{Q})$_being a solvable group, where $L$ is the splitting field of $f$. The group_$G$_acts faithfully_by_permuting_the roots of $f$ and_thus embeds in $S_d$ for_$d =_\deg f$. The groups $S_1, \dots, S_4$ are_solvable,_but $S_5$ is_not. We thus only need to write down some_$f$ with $G = S_5$, which_can be done_with_a_bit of computation; $f(x)_= x^5 - x + 1$_works, I think.
The argument above is_the basic proof you'd see in any_first Galois theory class, although the_original proof preceded Galois by_a decade_or so. Here's what looks_like a description_of that_proof, though I've_only skimmed it myself. There are_more modern proofs;_Arnol'd, for example, has one with_a_topological bent, but_it_boils_down (in_more concrete terms)_to_the fact_that_$S_5$ is not a solvable group.

_Q:

Why_devices that have same physical size and_resolution may have different_density?

I_am testing on Huawei_MediaPad T5 and Samsung Galaxy_Tab A.
They have the same _fiscal size_(10.1 inch)_and pixel number (1920*1200),
Question 1: Why they have different density (2.0_and 1.5)?  Density dipendes only_on fiscal size and_pixel number,_right?
Question_2: I gave_the_same width_to the sidebar (150dp) using dp, why_they look_different?
    <android.support.v7.widget.RecyclerView
 _   __ android:id="@+id/fcom_categories"
    _   android:layout_width="150dp"
  _    _android:layout_height="match_parent"
__    _ android:background="@color/white"
    _  _android:scrollbars="none"
      __app:layoutManager="android.support.v7.widget.LinearLayout