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i2 = −1.1 Gauss• 2, 3, 4 .• 1 + i, 2 − 3i .
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5 + 5i = (1 + 2i)(3 − i).
• 5 1 − 2i .5 = (1 − 2i)(1 + 2i).
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Gauss3 Gauss ε
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(a2 + b2)(c2 + d2) = 1.
a, b, c, d , a = ±1, b = 0a = 0, b = ±1.
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Gauss4 Gauss π
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1 Gauss α π1, . . . , πn
α = πm1
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• 4
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4 = 2 × 2 = −2i × 2i
• 5 + 5i
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5 + 5i = (1 + 2i)(1 − 2i)(1 + i)
= (2 − i)(1 − 2i)(−1 + i)
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Pythagoras (1/4)2 x, y, z x2 + y2 = z2 , x, y�
, u, v
x = u2 − v2, y = 2uv, z = u2 + v2
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x2 + y2 = z2 ⇔ (x + iy)(x − iy) = z2
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Pythagoras (2/4)1 x ± yi .
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Pythagoras (3/4)x ± yi . �
(x + yi)(x − yi) = z2
x + yi
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x + yi = ε(u + vi)2, ε = ±1,±i, u, v .
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x = ±(u2 − v2), y = ±2uv, z = ±(u2 + v2)
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Pythagoras (4/4), .
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1 f(x), g(x), h(x) �
f(x), g(x)
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f(x)2 − g(x)2 = h(x)2
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u(x), v(x) ,
f(x) = u(x)2 + v(x)2, g(x) = 2u(x)v(x),
h(x) = u(x)2 − v(x)2
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(1/5)Fermat 1601 ,
• 1438 , .• 1453 ,
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(2/5)
x3 + ax2 + bx + c = 0
x = t − a/3� �
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t3 + dx + e = 0
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d = b −a2
3, e = c −
ab
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27
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(3/5)1 ω = −1 +
√3i/2
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ω2 + ω + 1 = 0
ω3 = 1
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t3 − 3uvt − u3 − v3
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(4/5)
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−3uv = d, −u3 − v3 = e
u, v ,
t = u + v, ωu + ω2v, ω2u + ωv
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u3v3 = −d3
27, u3 + v3 = −e
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(5/5)u3, v3
s2 + es −d3
27= 0
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2
x3 − 6x2 + 11x − 6 = 0
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– – p. 19