gf(2) Find polynomial factors

Find all the irreducible factors of f

f = b0000110010101110101000111111010001111111111000000000000000110011
f = 0x0caea3f47fe00033
f = x59 + x58 + x55 + x53 + x51 + x50 + x49 + x47 + x45 + x41 + x40 + x39 + x38 + x37 + x36 + x34 + x30 + x29 + x28 + x27 + x26 + x25 + x24 + x23 + x22 + x21 + x5 + x4 + x + 1

Factor out any squareable factors in f.

f = (b0000000000000000000000000000000000000000000000000000000001100111)2(b0000000000000000111101111101100100100100100111111111001100111111)
f = (0x0000000000000067)2(0x0000f7d9249ff33f)
f = (x6 + x5 + x2 + x + 1)2(x47 + x46 + x45 + x44 + x42 + x41 + x40 + x39 + x38 + x36 + x35 + x32 + x29 + x26 + x23 + x20 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x12 + x9 + x8 + x5 + x4 + x3 + x2 + x + 1)

For each factor found, factor into irreducible polynomials.

f0 = b0000000000000000000000000000000000000000000000000000000001100111
f0 = 0x0000000000000067
f0 = x6 + x5 + x2 + x + 1

Find irreducible factors of f0

f00 = b0000000000000000000000000000000000000000000000000000000001100111
f00 = 0x0000000000000067
f00 = x6 + x5 + x2 + x + 1

f1 = b0000000000000000111101111101100100100100100111111111001100111111
f1 = 0x0000f7d9249ff33f
f1 = x47 + x46 + x45 + x44 + x42 + x41 + x40 + x39 + x38 + x36 + x35 + x32 + x29 + x26 + x23 + x20 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x12 + x9 + x8 + x5 + x4 + x3 + x2 + x + 1

Find irreducible factors of f1

f10 = b0000000000000000000000000000000000000000001001111000101010011001
f10 = 0x0000000000278a99
f10 = x21 + x18 + x17 + x16 + x15 + x11 + x9 + x7 + x4 + x3 + 1

f11 = b0000000000000000000000000000000000000000000000000000000011000001
f11 = 0x00000000000000c1
f11 = x7 + x6 + 1

f12 = b0000000000000000000000000000000000000000000000000000000000000011
f12 = 0x0000000000000003
f12 = x + 1

f13 = b0000000000000000000000000000000000000000000000000000000101110001
f13 = 0x0000000000000171
f13 = x8 + x6 + x5 + x4 + 1

f14 = b0000000000000000000000000000000000000000000000000000000000110111
f14 = 0x0000000000000037
f14 = x5 + x4 + x2 + x + 1

f15 = b0000000000000000000000000000000000000000000000000000000000101111
f15 = 0x000000000000002f
f15 = x5 + x3 + x2 + x + 1

f factored, powers expanded:

f = (b1100111)(b1100111)(b1001111000101010011001)(b11000001)(b11)(b101110001)(b110111)(b101111)
f = (0x67)(0x67)(0x278a99)(0xc1)(0x3)(0x171)(0x37)(0x2f)
f = (x6 + x5 + x2 + x + 1)(x6 + x5 + x2 + x + 1)(x21 + x18 + x17 + x16 + x15 + x11 + x9 + x7 + x4 + x3 + 1)(x7 + x6 + 1)(x + 1)(x8 + x6 + x5 + x4 + 1)(x5 + x4 + x2 + x + 1)(x5 + x3 + x2 + x + 1)