Definition df-sfl1 35590

Description: Temporary construction for the splitting field of a polynomial. The inputs are a field 𝑟 and a polynomial 𝑝 that we want to split, along with a tuple 𝑗 in the same format as the output. The output is a tuple ⟨𝑆, 𝐹⟩ where 𝑆 is the splitting field and 𝐹 is an injective homomorphism from the original field 𝑟.

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
df-sfl1	⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))

Distinct variable group:   𝑓,𝑏,𝑔,ℎ,𝑗,𝑚,𝑝,𝑟,𝑠,𝑡

Detailed syntax breakdown of Definition df-sfl1

Step	Hyp	Ref	Expression
1		csf1 35577	. 2 class splitFld1
2		vr	. . 3 setvar 𝑟
3		vj	. . 3 setvar 𝑗
4		cvv 3464	. . 3 class V
5		vp	. . . 4 setvar 𝑝
6	2	cv 1538	. . . . 5 class 𝑟
7		cpl1 22145	. . . . 5 class Poly1
8	6, 7	cfv 6542	. . . 4 class (Poly1‘𝑟)
9		c1 11139	. . . . . . 7 class 1
10	5	cv 1538	. . . . . . . 8 class 𝑝
11		cdg1 26048	. . . . . . . 8 class deg1
12	6, 10, 11	co 7414	. . . . . . 7 class (𝑟deg1𝑝)
13		cfz 13530	. . . . . . 7 class ...
14	9, 12, 13	co 7414	. . . . . 6 class (1...(𝑟deg1𝑝))
15		ccrd 9958	. . . . . 6 class card
16	14, 15	cfv 6542	. . . . 5 class (card‘(1...(𝑟deg1𝑝)))
17		vs	. . . . . . 7 setvar 𝑠
18		vf	. . . . . . 7 setvar 𝑓
19		vm	. . . . . . . 8 setvar 𝑚
20	17	cv 1538	. . . . . . . . 9 class 𝑠
21	20, 7	cfv 6542	. . . . . . . 8 class (Poly1‘𝑠)
22		vb	. . . . . . . . 9 setvar 𝑏
23		vg	. . . . . . . . . . . . 13 setvar 𝑔
24	23	cv 1538	. . . . . . . . . . . 12 class 𝑔
25	18	cv 1538	. . . . . . . . . . . . 13 class 𝑓
26	10, 25	ccom 5671	. . . . . . . . . . . 12 class (𝑝 ∘ 𝑓)
27	19	cv 1538	. . . . . . . . . . . . 13 class 𝑚
28		cdsr 20327	. . . . . . . . . . . . 13 class ∥r
29	27, 28	cfv 6542	. . . . . . . . . . . 12 class (∥r‘𝑚)
30	24, 26, 29	wbr 5125	. . . . . . . . . . 11 wff 𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓)
31	20, 24, 11	co 7414	. . . . . . . . . . . 12 class (𝑠deg1𝑔)
32		clt 11278	. . . . . . . . . . . 12 class <
33	9, 31, 32	wbr 5125	. . . . . . . . . . 11 wff 1 < (𝑠deg1𝑔)
34	30, 33	wa 395	. . . . . . . . . 10 wff (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))
35		cmn1 26120	. . . . . . . . . . . 12 class Monic1p
36	20, 35	cfv 6542	. . . . . . . . . . 11 class (Monic1p‘𝑠)
37		cir 20329	. . . . . . . . . . . 12 class Irred
38	27, 37	cfv 6542	. . . . . . . . . . 11 class (Irred‘𝑚)
39	36, 38	cin 3932	. . . . . . . . . 10 class ((Monic1p‘𝑠) ∩ (Irred‘𝑚))
40	34, 23, 39	crab 3420	. . . . . . . . 9 class {𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))}
41		c0g 17460	. . . . . . . . . . . . 13 class 0g
42	27, 41	cfv 6542	. . . . . . . . . . . 12 class (0g‘𝑚)
43	26, 42	wceq 1539	. . . . . . . . . . 11 wff (𝑝 ∘ 𝑓) = (0g‘𝑚)
44	22	cv 1538	. . . . . . . . . . . 12 class 𝑏
45		c0 4315	. . . . . . . . . . . 12 class ∅
46	44, 45	wceq 1539	. . . . . . . . . . 11 wff 𝑏 = ∅
47	43, 46	wo 847	. . . . . . . . . 10 wff ((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅)
48	20, 25	cop 4614	. . . . . . . . . 10 class ⟨𝑠, 𝑓⟩
49		vh	. . . . . . . . . . 11 setvar ℎ
50		cglb 18331	. . . . . . . . . . . 12 class glb
51	44, 50	cfv 6542	. . . . . . . . . . 11 class (glb‘𝑏)
52		vt	. . . . . . . . . . . 12 setvar 𝑡
53	49	cv 1538	. . . . . . . . . . . . 13 class ℎ
54		cpfl 35576	. . . . . . . . . . . . 13 class polyFld
55	20, 53, 54	co 7414	. . . . . . . . . . . 12 class (𝑠 polyFld ℎ)
56	52	cv 1538	. . . . . . . . . . . . . 14 class 𝑡
57		c1st 7995	. . . . . . . . . . . . . 14 class 1st
58	56, 57	cfv 6542	. . . . . . . . . . . . 13 class (1st ‘𝑡)
59		c2nd 7996	. . . . . . . . . . . . . . 15 class 2nd
60	56, 59	cfv 6542	. . . . . . . . . . . . . 14 class (2nd ‘𝑡)
61	25, 60	ccom 5671	. . . . . . . . . . . . 13 class (𝑓 ∘ (2nd ‘𝑡))
62	58, 61	cop 4614	. . . . . . . . . . . 12 class ⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
63	52, 55, 62	csb 3881	. . . . . . . . . . 11 class ⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
64	49, 51, 63	csb 3881	. . . . . . . . . 10 class ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
65	47, 48, 64	cif 4507	. . . . . . . . 9 class if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
66	22, 40, 65	csb 3881	. . . . . . . 8 class ⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
67	19, 21, 66	csb 3881	. . . . . . 7 class ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
68	17, 18, 4, 4, 67	cmpo 7416	. . . . . 6 class (𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩))
69	3	cv 1538	. . . . . 6 class 𝑗
70	68, 69	crdg 8432	. . . . 5 class rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)
71	16, 70	cfv 6542	. . . 4 class (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))
72	5, 8, 71	cmpt 5207	. . 3 class (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝)))))
73	2, 3, 4, 4, 72	cmpo 7416	. 2 class (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))
74	1, 73	wceq 1539	1 wff splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))

This definition is referenced by: (None)