polynomial equation root problem. It was not until 1830 was solved by galova , which belongs to a big leak in the history of mathematics.
As early as 1550, Italian mathematicians Cardano and Taltalia gave the root formula for cubic equation : Cardano formula . The right to invent the formula
is controversial between them and , so let’s mention their names here. After
, many great mathematicians have studied the problem of the root solution of the 5th equation, including: Gaussian , Lagrangian , , Caoci , Rufini, Abel .
Gaussian (1777-1855) has been very close to solving this problem.
He solved the problem of 17 17 2 ft gauges drawing .
The ruler drawing of the regular 17-sided edge is equivalent to making a complex number, that is, finding the 17 equal points of the center angle , that is, solving the algebraic equation
The drawing of the regular polygon is equivalent to the root of the equation
Of course, Gaussian research field There are many , such as the famous Euclidean fifth public problem, which is solved by Gaussian.
Whether it is calculus, differential geometry, or abstract algebra , it has many contributions from Gauss.
However, in the problem of the root solution of the 5th equation, Gauss does "A wise man will have a mistake," . The reason why
does this happen is that I think it is because Galova is from Viological , and he does not have the Mortal Mathematics Education 's thinking limitations brought by .
Galova 16 years old started to learn mathematics, and ordinary children have been learning math for 8-10 years at this age. Their thinking has been affected by teachers so much that it is difficult for to rethink what add, subtract, multiply, divide, and preview is.
However, a civil subject who only started to learn mathematics at the age of 16, he did not have the enlightenment teacher , and naturally he also had no thinking limits .
So, it may be easier for him to solve the complex problem of rooting the equation, and to solve the complex problem of recursively step by step [grin]
If the rooting problem of equation is considered from the perspective of programming, then since 5 operators are too complicated, we will first study 1 of [cover the face]
After Galois did this, indeed looked at in hindsight, but even Gauss did not think of it at that time.
Garois once gave the paper to Cauchy , but Cauchy lost it, and Cauchy's careless was also warned by Lagrangian before his death.
Lagranges is Cauchy's teacher. He always feels that he is not careful enough, and it is simply a prediction in advance of .
Later, Garova gave the paper to Fourier , but Fourier just passed away . This paper is simply a death-threatening [covering your face]
Finally, Garova gave the paper to Liu Weier and Gauss , and specifically asked Gauss to publish it for him, but in the end it was Liu Weier to publish it for him.
or above is a story between several great mathematicians.
Next, let’s start with 1 operators .
1, the set and its binary operator,
set and the binary operator defined on it, if the three axioms of binding law, unit element, and inverse element are satisfied, it is called group :
1)
2)
3)
This operator is usually called "multiple" : it can be a multiplication of numbers in a narrow sense, or it can be "multiple" in a broad sense.
For example, the rotation of the clock pointer can also be regarded as an multiplication : it is essentially a multiplication of complex numbers, if you use Euler formula .
If it also conforms to the exchange law , it can be called "addition" , and this group is also called exchange group and Abel group .
Multiplication Usually is not swapped, but integer multiplication is swapped.
2, exchange law is very important,
4) ab = ba, the function of
exchange law when solving equations in is that it can synthesize into the process of coefficients, and then disassemble back to .
ax = b Solution: x = 1/a b. If it does not conform to the commutation law, it cannot be written as b 1/a or b/a.
because the definition of the group only stipulates that a has inverse element (reciprocal number), but it does not stipulate that "multipliation" is exchanged by .
So, since a is on the left of x, you can only multiply 1/a on the left, not 1/a on the right [covering your face]
integer can be done, but if the matrix equation AX = b, then X = A^-1 b and b A^-1 are different of , and the latter may even do not meet the multiplication rules of matrix.
However, there is an element multiplication in the group that definitely satisfies the commutational law, that is, unit element : see Axioms 2 and 3 in Section 1.
3, the group is not necessarily the addition of integer or multiplication , but only needs to meet the 3 conditions in Section 1 of .
So by changing 1, 2, 3, 4 into a substitution of 2, 3, 4, 1, it can also be a group:
1 -- 2,
2 -- 3,
3 -- 4,
4 -- 1,
can be abbreviated as (1, 2, 3, 4), using C code is y = x % 4 + 1. A group like
is called loop group .
Broadly speaking, there can be various groups.
4, group isomorphism,
In order to classify these various groups, a correspondence relationship is defined.
is the same as the set corresponding to natural number , which is called countable set . The two groups are also defined by the correspondence relationship .
If there are 2 groups G and G', the multiplication above them is x and * respectively. If there is an correspondence relationship f:G--G', satisfying:
1) f(axb) = f(a)*f(b),
2) f is a double shot, that is, corresponds to relationship one by one.
is true for any a and b belonging to G, so they are isomorphic .
For example, 1-26 corresponds to a-z one by one, and a corresponding relationship f can be defined, stipulating:
f(2x3) = f(2)xf(3)= ('b' - 'a' + 1)*('c' - 'a' + 1),
is calculated in this way on C language , because ASCII from a-z is continuous.
is exactly the multiplication of when is written by hand, and * when is multiplication of when programming by , because x represents the 24th English letter: x.
When converting hexadecimal to decimal, it is estimated that many people have written the transformation between a-f and 10-15. Its principle is the isomorphism of group [laughs]
If only meets the first , and does not meet the second (one-to-one correspondence), then it is the homomorphic of the group.
If isomorphic , then the unit element e' of G' will correspond to the unit element e of to G.
If is homomorphic , then the unit element e' of G' may correspond to multiple elements in G:
For example, x^2 = 1, then x can be 1 or -1. These multiple elements in
G that can be mapped by f to e' are called homomorphic kernel: ker f.
Its value lies in that f(ag) = f(a)*f(g)=f(a)*e'=e'*f(a)=f(g)*f(a)=f(ga), that is, it is that conforms to the exchange law .
That is to say, under "homomorphic map f" :
a (Ker f) = (Ker f) a, (Formula 1)
has done this to see what the situation is in exchange law can be used in .
Multiply to the right of of formula 1 by the inverse element of a, and you can get:
Ker f is a subgroup of group G, called regular subgroup .
5, regular subgroup is aK = Ka, and the commutative law can be used.
Take integer as an example. K = {1} is "regular subgroup" . If a = 2, then 2K is the set of all even . The example of
integer is useless, take a look at this example:
Transformation relationship between numbers
Assuming that a group consists of correspondence between 1, 2, 3, 4:
f: 1 - 2, 2 -- 3, 3 -- 4, 4 -- 1,
g: 1 -- (2) - 3, 2 -- (3) - 4, 3 -- (4) - 1, 4 -- (1) -- 2,
h: 1 -- (2,3) -- 4, 2 -- (3,4) -- 1, 3 -- (4,1) --2, 4 --(1,2) -- 3,
e: 1 -- (2,3,4) --1, 2--(3,4,1) --2, 3 --(4,1,2) --3, 4 --(1,2,3) --4.
G = {f, g, h, e}, consists of 4 elements, all of which correspond to 1234.
where e is unit , because of its effect on numbers, does not change the number .
So, e is a regular subgroup of G, fe(1) = f (e(1)) = f(1) = 2, ef(1) = e(f(1)) = e(2) = 2, so fe = ef, that is, this is that conforms to the commutation law .
, the transformation of numbers can be used on the subscript of several root of the root of the .
g is actually obtained by using f to twice in succession. It can be considered that g is to 2 power of f :
g = f(f(1, 2, 3, 4)) = (3, 4, 1, 2), the numeric order of in brackets on both sides of the equal sign is the corresponding relationship .
Similarly, h is 3th power of f, and e is at 4th power of f.
This group generated by using of f's multiple times ( until it does not work ) is called loop group . The effect of
f and g on the number 1234. Can be interchanged with ?
Some groups have elements that can be interchanged, while others cannot be interchanged, which leads to the concept of a subgroup of positions that can be exchanged.
6, Transfer subgroup,
fg = gf, this equation does not necessarily hold.
but it must be true: because after it is simplified, it is fg = fg.
So, the four formulas in brackets are called change seat : that is, operator required to exchange 2 elements .
In short, everything is for exchange law , because there is no provision in the definition of the group that the exchange law must be established.
As long as can use the commutation law, how can it be transformed into ?
But if the exchange law cannot be used, is finished [cover face]
Unit element e and other elements must be interchangeable . Unit element e can also form a subgroup of group G: because exe = e = exe, the combination law, unit element, and inverse element are all satisfied.
G If in addition to e, there is a larger transposition subgroup (G1) of (), it would be even better.
If its seat-transfer subgroup G1 also has a seat-transfer subgroup G2 that is larger than e, then it can form a chain : G - G1 - G2 - e.
As long as a group of transit subgroups can be reduced to unit e along this chain, then it is that can solve group .
That is to say, can solve all the functions of elements in group on numeric sorting , which is reversible ! How to disrupt the number of
, and how can I recover it back [grin]
Recovery steps required to restore , which is the length of delinkable .
So the solution to one-element equation ax + b = 0 requires up to 2 steps: consider rational numbers Q to Its own corresponds to group of composed of :
1, an element f in G, the effect on other numbers remains unchanged, but only ax is changed to -b,
2, and another element g in G, the effect on other numbers remains unchanged, but only x is changed to -b/a,
According to mathematical convention, change the two words "changed" to with = .
7, can be decomposed, single group,
single group, not decomposed.
single group transpose subgroup is equal to itself, cannot construct can be unlinked.
PS: The element of the group is not necessarily the number , but can also be the correspondence between .
Abstract algebra is the same as functional analysis . The element of the group can be " function " , and the element in the space can also be "function" .
What is the function ? corresponds to .
Multiplication ? function relationship [cover your face]
What is the functional? distance . What is the draw of
? multiplication [laughing cry]
distance definition
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