Metamath Proof Explorer


Definition df-gf

Description: Define the Galois finite field of order p ^ n . (Contributed by Mario Carneiro, 2-Dec-2014)

Ref Expression
Assertion df-gf GF = ( 𝑝 ∈ ℙ , 𝑛 ∈ ℕ ↦ ( ℤ/nℤ ‘ 𝑝 ) / 𝑟 ( 1st ‘ ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cgf GF
1 vp 𝑝
2 cprime
3 vn 𝑛
4 cn
5 czn ℤ/n
6 1 cv 𝑝
7 6 5 cfv ( ℤ/nℤ ‘ 𝑝 )
8 vr 𝑟
9 c1st 1st
10 8 cv 𝑟
11 csf splitFld
12 cpl1 Poly1
13 10 12 cfv ( Poly1𝑟 )
14 vs 𝑠
15 cv1 var1
16 10 15 cfv ( var1𝑟 )
17 vx 𝑥
18 cexp
19 3 cv 𝑛
20 6 19 18 co ( 𝑝𝑛 )
21 cmg .g
22 cmgp mulGrp
23 14 cv 𝑠
24 23 22 cfv ( mulGrp ‘ 𝑠 )
25 24 21 cfv ( .g ‘ ( mulGrp ‘ 𝑠 ) )
26 17 cv 𝑥
27 20 26 25 co ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 )
28 csg -g
29 23 28 cfv ( -g𝑠 )
30 27 26 29 co ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 )
31 17 16 30 csb ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 )
32 14 13 31 csb ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 )
33 32 csn { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) }
34 10 33 11 co ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } )
35 34 9 cfv ( 1st ‘ ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } ) )
36 8 7 35 csb ( ℤ/nℤ ‘ 𝑝 ) / 𝑟 ( 1st ‘ ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } ) )
37 1 3 2 4 36 cmpo ( 𝑝 ∈ ℙ , 𝑛 ∈ ℕ ↦ ( ℤ/nℤ ‘ 𝑝 ) / 𝑟 ( 1st ‘ ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } ) ) )
38 0 37 wceq GF = ( 𝑝 ∈ ℙ , 𝑛 ∈ ℕ ↦ ( ℤ/nℤ ‘ 𝑝 ) / 𝑟 ( 1st ‘ ( 𝑟 splitFld { ( Poly1𝑟 ) / 𝑠 ( var1𝑟 ) / 𝑥 ( ( ( 𝑝𝑛 ) ( .g ‘ ( mulGrp ‘ 𝑠 ) ) 𝑥 ) ( -g𝑠 ) 𝑥 ) } ) ) )