The Impossibility of Trisecting the Angle and Doubling the Cube (metamath 100 #8)

[tirix]

An elegant proof would certainly rely on Galois Theory.

Such a proof is provided Chapter 7.4 of [Ian Stweart - Galois Theory], p.99, theorems 7.13 and 7.14.

[savask]

Maybe worth mentioning that the proof doesn't require a great deal of Galois theory, in fact the only crucial bit seems to be that field extensions are multiplicative, and that is already proved in set.mm in extdgmul. The fact that the cubic root of 2 gives us a field extension of degree 3 can be proved by elementary means, by mimicking the argument about minimal polynomials.

Probably the most laborious part would be to define constructible numbers and prove that they give field extensions of degree a power of 2. I took a look at how other formalization systems do this, and both HOL Light and Isabelle go with the geometrical definition. That is, we start with two points (if we identify points on the plane with vectors, then usually (0, 0) and (1, 0)) and call them constructible by default. Then:

1. if points A, B, C, D are constructible, then the intersection of proper non-parallel lines AB and CD is again constructible,
2. if points A, B, C, D, E, F are constructible, then the intersection points of a proper line AB with a circle with center at C and radius |EF| are again constructible,
3. finally, if A, B, C, D are constructible, then the intersection points of two distinct circles with centers A, C and radii |AB|, |CD|, respectively, are also constructible.

I think it would be beneficial to model constructible numbers in set.mm as a subset of complex numbers, as we do in section "Theorems of Pythagoras, isosceles triangles, and intersecting chords" of set.mm. Geometry of CC is more developed in set.mm than the Tarskian geometry, and this gives an additional benefit that we may identify points on the plane with points of CC and hence a real number is constructible if the corresponding complex number (i.e. point) is constructible.

I unfortunately don't know how recursive definitions work in set.mm, so I cannot draft a preliminary definition of constructible (complex) numbers, but it seems that we should start with 0 and 1 as being constructible by default and properties 1)-3) can be easily stated with standard operations with complex numbers, for example, one can use something in the spirit of sigarcol to test if points lie on the same line.

The next step would be to show that construcible numbers form a field (again, I think the fact that we would be able to use + and x on complex numbers should simplify the proof), and then perhaps the most difficult step would be to show that if a number is constructible then the corresponding field extension degree is a power of 2. I don't know if set.mm has notation for a field generated by a subfield and some element, but more importantly, we would have to prove that intersections of circles and lines require only quadratic field extensions.

I would be glad to participate and discuss the details if someone is willing to try formalizing some of the definitions and intermediate steps required.

[icecream17]

Something like ( rec ( ( s e. _V |-> AllPointsConstructibleFrom_s ) , { <. 0 , 0 >. , <. 1 , 0 >. } ) " _om ) would work

[savask]

I tried giving a definition of constructible points (and numbers) using icecream17's rec idea:

$c Constr $.
cconstr $a class Constr $.

df-constr $a |- Constr = ( rec ( ( s e. _V |-> { x e. CC |
 ( ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ -. ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) = 0 ) \/
 E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) ) \/
 E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( -. a = d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) )
 } ) , { 0 , 1 } ) " _om ) $.

Here constructible points are understood as a subset of complex numbers CC. We start with 0 and 1 being constructible by default, and the first line says that the intersection of two not-parallel constructible lines is constructible (notice how the expression with Im checks for collinearity, it was used in sigarcol for the same purpose), second line is for the intersection of a line and a circle, and the third line is for the intersection of two circles with different centers. This is pretty much the same definition as Isabelle or HOL Light use, but inside the complex plane.

I guess one could similarly give a more general definition of "points constructible from some set", but I don't think this is important enough to complicate the definition.

What do you think? Does this definition look right? If we agree on this definition, we could try writing down intermediate statements required to prove the metamath 100 question. Norman suggested formalizing statements (without proofs) of metamath 100 theorems in the past, so this could be a good idea.

[tirix]

Nice initiative! It was not immediately clear to me, but the actual condition to express is that the intersecting lines and circles should not be the same. Otherwise the whole line/circle gets added, and we end up with too many constructible points. If there is no intersection, no point gets added. Your condition expressed with `|Im|` and the `|-. a = d|` condition are both sufficient conditions for the line, resp. circle to be distinct, and it does not require an additional bound variable, so it looks correct, and clever! I don't think we have a notation yet for a field generated by a subfield and and some additional element (like e.g. ℚ[√2]), that would certainly be a nice addition, and I'd be interested to contribute!

…

On 07/01/2025 08:41, savask wrote: I tried giving a definition of constructible points (and numbers) using icecream17's |rec| idea: |$c Constr $. cconstr $a class Constr $. df-constr $a |- Constr = ( rec ( ( s e. _V |-> { x e. CC | ( ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ -. ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) = 0 ) \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) ) \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( -. a = d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) } ) , { 0 , 1 } ) " _om ) $. | Message ID: ***@***.***>

[savask]

so it looks correct

Nice, so at very least we can express the statement of the theorem now (cubic root of 2 and cos(pi/9) don't lie in Constr).

I don't think we have a notation yet for a field generated by a subfield and and some additional element (like e.g. ℚ[√2]), that would certainly be a nice addition, and I'd be interested to contribute!

That's an interesting question whether we really need a full-fledged definition for the proof. We certainly want to work inside CC, so let's say s is a subfield of CC (viewed as a subset), and c e. CC is such that ( c x. c ) e. s, so a square root of some element from s. Then

{ x e. CC | E. a e. s E. b e. s x = ( a + ( b x. c ) ) }

is also a subfield of CC. We want to prove that any constuctible number can be obtained by a finite series of field extensions like that. Maybe we could define "algebraically constructible numbers" as

$c ConstrAlg $.
cconstralg $a class ConstrAlg $.

df-constralg $a |- ConstrAlg = ( rec ( ( s e. _V |-> { x e. CC | E. a e. s E. b e. s E. c e. CC ( ( c x. c ) e. s /\ x = ( a + ( b x. c ) ) ) } ) , QQ ) " _om ) $.

and then we need to prove that Constr C_ ContrAlg (they are equal, of course, but we require only inclusion for impossibility proofs). Now, I'm not sure if it's straightforward enough to relate quadratic field extensions inside CC I described earlier with this definition of ConstrAlg, so your thoughts are appreciated. Certainly the recursive operation in the definition of ConstrAlg doesn't always give a field, so one would need some sort of proof by induction, showing that if a number can be obtained from QQ by k iterations of this operation, then it lies in a finite field extension of QQ of index a power of 2.

[tirix]

I believe this approach would work, and we don't really need the full-fledged definition of a generated field for the proof. But it would be sooo nice to use this opportunity to formalize more of the Galois Theory and have some basic bricks for e.g. the insolvability of the quintic...

[tirix]

My thought was defining something like:

df-fldgen $a |- fldGen = ( f e. _V |-> ( e e. ~P ( Base ` f ), s e. ~P ( Base ` f ) |-> |^| { a | ( s C_ a /\ f /FldExt a /\ e /FldExt a ) } ) )

Then given a universe field F, a subfield E and a generating set S, ( E ( fldGen ` F ) S ) would be the field extention of E generated by S in F. E.g. we could write ( ( CCfld |`s QQ ) ( fldGen ` CCfld ) { ( sqrt ` 2 ) } ).

[savask]

But it would be sooo nice to use this opportunity to formalize more of the Galois Theory and have some basic bricks for e.g. the insolvability of the quintic...

I agree, then let's see if we can formulate it in Galois theory terms.

My thought was defining something like: ...

Any reason why we need a subfield E at all? Why not define fldGen for a universe field F and a subset S, so F fldGen S will be the field generated by S inside F. In theorems where we extend a subfield E with elements from S we could use an idiom F fldGen ( ( Base ` E ) u. S ).

Also, in your definition you intersect fields with |^| operation, is it going to work? I don't understand extensible structures in set.mm very well, but I would assume that one first has to intersect sets of the form ( Base ` a ), and then repackage the result back into a structure.

[tirix]

In your definition you intersect fields with |^| operation, is it going to work? I don't understand extensible structures in set.mm very well, but I would assume that one first has to intersect sets of the form ( Base ` a ), and then repackage the result back into a structure.

You're right, this would not work. For an intersection-based definition, the resulting object has to be a subset of the base. We can then recover a field by using the structure restriction operator: ( F |`s X )

Any reason why we need a subfield E at all? Why not define fldGen for a universe field F and a subset S, so F fldGen S will be the field generated by S inside F. In theorems where we extend a subfield E with elements from S we could use an idiom F fldGen ( ( Base ` E ) u. S ).

That's a good remark, this would simplify the definition to really "a field generated by a set" without even mentioning field extension.
A definition could be:

df-fldgen $a |- fldGen = ( f e. _V , s e. ~P ( Base ` f )  |-> |^| { a e. ( ( SubRing ` f ) i^i Field ) | s C_ a } ) )

This resembles the definition of sigma algebra generated by a set (df-sigagen), or that of the ring spanned by a set (df-rgspn)

[tirix]

See #4553 for my concrete proposal.

[tirix]

I think another thing we would need to formalize is the idea of "tower" of field extensions. Ideally, we should find a way that is generic enough so that it could also be used for towers of normal subgroups, as these are useful for the definition of solvable groups.

I believe the best is to see such towers as chains, i.e. see the subfield (resp. normal subgroup) relation as an order relation, so that the chains are subsets where the order is total.

We already have definitions for Posets and Tosets, so I think we could prove some generic facts in the settings of order theory, and use them for such chains (I don't think we need a lot of them)

[tirix]

I believe the best is to see such towers as chains, i.e. see the subfield (resp. normal subgroup) relation as an order relation, so that the chains are subsets where the order is total.

(That's actually what I had in mind when I defined the Field Extension as a relation)