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Fully Implement the algorithm in the Sage computational algebra system. A separate approach which we have investigated involves first factoring f_a(x) over Q_p. Develop and compare the relative effectiveness of this second algorithm. Note: This would require formulating and proving a polynomial factorization version of Hensel s Lemma. References: Frhlich A, Taylor MJ. Algebraic Number Theory. Cambridge University Press; 1991. Wikipedia contributors. "Newton polygon." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 6 Apr. 2026. Web. 16 Apr. 2026. Acknowledgements: McNair Scholars Program Dr. Sandra Suarez, Ramapo McNair Program Director Prof. Ken McMurdy, Project Supervisor L p   cWa  $$ (( ,,0044O88 < n=2, m=3, e=1 p=13: (x^2 + 5)(x^2 + x + 8)(x^2 + 12x + 8) -> n=3, m=2, e=1 p=17: (x^3 + 14x + 3)(x^3 + 14x + 14) -> n=2, m=3, e=1 p=19: (x^2 + 4)(x^2 + 9x + 7)(x^2 + 10x + 7) -> n=3, m=2, e=1 p=29: (x + 6)(x + 8)(x + 14)(x + 15)(x + 21)(x + 23) -> n=6, m=1, e=1 Etc. These are the easy primes! 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Hensel s Lemma shows that f_a(x) has a root in Z_5 (and hence all six roots lie in Z_5). Here are their approximations: a_0 = 1+1*5+1*5^2+0*5^3+4*5^4+& a_1 = 1+4*5+1*5^2+3*5^3+4*5^4+& a_2 = 2+0*5+3*5^2+3*5^3+3*5^4+& a_3 = 3+4*5+1*5^2+1*5^3+1*5^4+& a_4 = 4+0*5+3*5^2+1*5^3+0*5^4+& a_5 = 4+3*5+3*5^2+4*5^3+0*5^4+& So, the prime p=5 completely splits in K, i.e., e=m=1 and n=6.Hc\ c "  $$((,,004488<< c "  $$((,,004488<<    $$((,,0c04"488<<    $$((,,004488<<  c"  $$((,,004488<<    $$((,,00448c8<"<    $$((,,004488<<  c"  $$((,,004488<<    $$((,,?00    bNNNNNNNNNNNNNNNNNNA  4 tl?Rectangle 14" PK!2<>[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!cIfXdrs/shapexml.xml]n6w t;:F"M@Vu{h(n6`vӷv)Y#4r{D&-Jw~)Y+ C0! ywXxK1tnD<'A2apLsdl&x( XE\CU7'J$BjBQZféhr侼:U,N8LJЇPvi!qIJg<ůK`2> Wkne=Acz׿tOu w3_E >A_V@e~bkw m:{.`ۉJ"޸g O4t-:ITv 0ܔ .Ci5Pek:aa簛:]p[9J͢@ : Xh9Ah@PlLlnw?wNmy^>yʳ;g;A%A9,|!ݝ^d.T6.(LH߀(ڰCd]@(|"N(2 F2H=<<!:k>4T]Jւg4c!c @x|7 KScWB9XH&-sD- m&|c|Btx c̐u<6敗c+tCK A]/QſH!fZ1xlll%Z d<2%] *4|,#%*0x+~1U$yʅ2>Uq웲( ½ed/Vm%2Y:A>Ң&ұr#-D܅p/J|ǧO?}݃ s"E&Q +eG^7%+JVvpeQij R2R4y4&F)W?;ƓD1lTjxIN4 ,.FY& &M]$e! K)8kzl-S+N+Wۚ5!6mAB#_/OBPvS_X- iRM- Wy>-X"% k4q Y̤DB"8M0e XmfgAT*`b66d"`8}f8HyBΟK _|Az|L K4)i҇%F=AS=KXg.: ߩ)S$Rٔ*;ܖ?>b:L MâcEh y߆÷^dQ΁eji̵ O*P֢HYEEӠ-">d a51՝"؊͓ HBUOkT':hP` 5䪦1"d]*-sO`6Y3@1gn~qz,jY2BW[cՄ%˃,QH,,~28mpOKvJ&Gz/Kf ujgil>̑Gp$7~h$X9HH]H"Xv=ySGa89˛g `;S7Zz ۴yy :6_=_3qg۴7_9SPB%~9w j I. s\qz%͗ o$v Bo]㭱ag<]˄RΪ,dPK!hVdrs/downrev.xmlDN0EH5HSBU!UBn'6MX,`yuYFӉ:0e ؖNFx{8"YM0qhAvak 6RaCaz7SԞ7TYKCM |p6骾ܟ_5j@<=׷ "|ѫE=k D4}Vo)DI/&QPK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!cIfX(drs/shapexml.xmlPK-!hV drs/downrev.xmlPK !p#G8 , For the primes p that do divide the discriminant, we must explicitly represent the roots of f_a(x) in K (equivalently the automorphisms in Gal(K/Q)). [a, -a, 1/30*a^4 - 1/6*a^2 + 1/2*a + 2/15, 1/30*a^4 - 1/6*a^2 - 1/2*a + 2/15, -1/30*a^4 + 1/6*a^2 + 1/2*a - 2/15, -1/30*a^4 + 1/6*a^2 - 1/2*a - 2/15] First, let s analyze the p-adic geometry of these roots for the prime p=3. Newton Polygon implies that v_3(a-1)=1 for three roots and v_3(a-2)=1 for three roots. We immediately see that e=1 and n=2 (and so m=3). In order to focus on the roots near z_0=1, we make the substitution b=(a-1)/3. (So, b is now a unit in O_K^*.) The reduction of b now satisfies the polynomial: x^3 + x^2 + 2 So, not only does this confirm that m=3, as this polynomial is irreducible over F_3 and hence generates F_{3^3}. It also provides an explicit representation from which Frob_3 can easily be determined. ^G c #c " $$(c(,",0c04"488V ^ L    4 A,??,Ramapologo-multicolorPicture 60Ramapologo-multicolor    4 HH?Rectangle 1  <"  Defining the Problem: Let K=Q(a) be a Galois number field (a an algebraic integer), and let f_a(x) be the (monic) minimal polynomial of a. Let O_K be the ring of integers in K (analogous to Z in Q). Theorem: For each prime p in Z, the ideal p O_K factors uniquely as p O_K = P (p_i)^e, i =1,..,n, where the p_i s are distinct prime ideals of O_K. Moreover, O_K/p_i is isomorphic to F_{p^m} for some m, and emn = [K:Q] = deg(f_a(x)). Theorem: Let G_p < Gal(K/Q) be the decomposition subgroup at p, consisting of those automorphisms s satisfying s (p_i)=p_i for all i. The aforementioned isomorphism of O_K/p_i with F_{p^m} induces a surjection of G_p onto Gal(F_{p^m}/F_p) = Z/mZ. Problem: Let Frob_p be the automorphism in Gal(F_{p^m}/F_p) defined by Frob_p(x)=x^p. Determine, using p-adic analytic methods, the automorphism s in G_p that maps onto Frob_p.r2  c c!c)  $c$@(( ,,C004c488[Content_Types].xmlAN0EH%,BM #{X$ccB{{&m TҞy?٫vՌ}Z_VH68O}6ōV;dnV]DVBz9v I&]Hd9DУkce\%C7;Dp{KU!["jK8 p&î8By/ O4;Tϐ#La\bDR7q[,į W pov+ fCPK!]  _rels/.relsj1 @h_B֐BWaLΖ5yB/dˠA 0xM e`J FG2p%}iRdITD10Z)4(I9`mtB{ƁmtπnTg 險,ܗr>j1W*@ŀӚZ7O#KOiPK!ZH drs/shapexml.xml]nH/0@:ز Fd` (9m9Iy33vmyq(ԭ|B"T꧛?^]Q'c u#}=߾ys':at_~Ly:(Q܄KITfGi.TA?N-Usӌ_'i|tqGen.ۚQ\K׋lYޢz-3wܝ'yzrv|LtON=}zoHubtbbDwpӳtߟnfy0<ϨwX]'_Lob}}R'm3yrU$wͅ'⼘$ Dg* oˉOײ9 NT5!!yxf'I´եn"K9J4u \Sӕ9z4(;EЪJjq.|rQvWZj5{_o?.M ueSo6P!XUcFzZFt)?OHuP}D3MV_Ofɢ[Nf{azHN ia5k8zMvNY,d{eAXk[M4 @̑<9~cRqQNp U}e2嶌OÅ ^fܨ,eH߆???}o>˓O"޻Շ"̓HtIJ`rC`GŽJ R $! w K,Z(!CjSQÚe<\/ķP2qoX$F24 NL|}`qYW3lHX1_^YT_yQrX󋫌HRPJ&:h$QV]4" p`.Y lȺHbm@I?krC#޻ t -Qn&x"-RIz MkPT ҂Ηk Z]uqI&1!+8I&]#dmC-ϒĦIEH:t#NnRč@Z<;rS.{'ēk"qπ6z],~ #c|A1lSKxeܠ /fbu& P!`5 W*x.b}> uTysPe.I:(K߅hVtc?,% dVnVXiQژ 4MjK?D~W.լbE';CL*L&&Is,ᅬ/XATr` z:QT^~J,R퇲/Ang:kh3+HUn|bB3UU9w@P\5P{q!2Nd/&$:P.RS(N<pե~kWr;VE[䃣GqSGQj$`*$BKa#*,}4>$RW[颷aR?mG@+dm>`(D>8͐%jE٭UtP29ĕT\U+^ jܩr@dN9cĐ]#Jwa1b6\gXDp 2$BHpwMQ4xpԸ8gemyeL~&<.VzۙߝMF_&zeE!dnLO=tz0YEhSw#1}3S6,pyso͙{Ӳ0`dK%춫 *l%  0M1Sl7t*bĆ %41vl`l* }͕s1M vh<Fz-r~8믧|?^J~1.W]%n+ kKHn4YPK-!2<>[Content_Types].xmlPK-!]  ,_rels/.relsPK-!ZH (drs/shapexml.xmlPK-!C drs/downrev.xmlPK E( ]I & Description of the Algorithm: Step 1: Compute the discriminant of f_a(x). For any prime p not dividing disc(f_a(x)), the factorization of p O_k exactly matches the factorization of the reduction of f_a(x) mod p. (These are the easy primes, and they are all but finitely many.) Step 2: For any prime that does divide the discriminant, determine (via Hensel s Lemma) if f_a(x) has a root in Z_p. Step 3a: If f_a(x) has a root in Z_p then G_p is trivial and e=m=1. The ideal p O_K splits completely, i.e., factors into the product of [K:Q] distinct primes. Frob_p is just the trivial automorphism in this case. Step 3b: On the other hand, if it does not, use the Newton Polygon to find z_0 in Z_p such that v_p(a-z_0) (p-adic valuation) is as large as possible. I.e., we find the nearest point to a in Z_p. Step 4: (continuing 3b) Make the substitution b=(a-z_0)/p^i where i=[v_p(a-z_0)]. This translates all G_p-conjugates of a into the residue disk of the p-adic completion of O_K centered at 0, and guarantees that v_p(b)<1 (as well as the G_p-conjugates b_i of b). Step 5: Use the Newton Polygon to compute v_p(b_i) and v_p(b_i-b_j). The ramification index e is the lcm of the denominators of these p-adic valuations. m is then forced to be |G_p|/e. 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