Taylor 1
Ashley Taylor
Dr. William Paulson
Math 6613-Abstract Algebra II
18 April 2012
The Three “Impossible” Constructions
The word impossible means: “ not possible; unable to be, exist, happen; unable to be
done, performed, effected; incapable of being true, as a rumor; not to be done, endured with any
degree of reason or propriety: an impossible situation.” (Dictionary.com) Now thinking about the
world of mathematics, one might hesitate to use this word because of its implied nature. If
something is proved to be “Impossible” then the word can be used with no doubt. When one
considers Euclid’s rules of compass and straightedge construction there are three “impossible”
constructions: “Trisecting the angle: Given an arbitrary angle, construct an angle exactly onethird as great; Duplicating the cube: given a cube of arbitrary volume, find a cube with exactly
twice the volume; Squaring the circle: given an arbitrary circle, find a square with the same
area.” (Suzuki 27-28) These three constructions have been proven impossible by two
mathematicians Pierre Laurent Wantzel and Ferdinand von Lindemann using Galois Theory of
fields and the fact that π is a transcendental number; it is not a root of any polynomial with
rational coefficients.
To give some background on the mathematicians who have made a tremendous
contribution to these “Impossible” constructions in mathematics. First there is Pierre Laurent
Taylor 2
Wantzel was responsible for proving the that duplicating a cube and trisecting an angle were
impossible to construct using only a compass and straight edge.
“He was born on June 5, 1814 in Paris, France and at the age of nine his teachers would
send for him to help with certain difficult surveying problems. He attended college at the
College Charlesmagne starting in 1828. At 15, Wantzel edited ‘Antoine-Andre-Louis
Reynaud’s’ Treatise on Arithmetic where he added a proof of finding square roots. It
was not until 1837 when he became the first to prove the impossibility of duplicating the
cube and trisecting an angle.” (Freeman)
This is a little bit on Pierre Lauren Wantzel. Now let’s look at German mathematician
Ferdinand von Lindemann who was
“born April 12, 1852 in Hannover, Germany. He studied at the following schools:
University of Gottingen, University of Munich, and the University of Erlangen where he
received hid doctorate in 1873. He later taught at the University of Freiburg from 1877
to 1883. His proof of the transcendence of pi was published in 1882 in a article called
‘Concerning the number pi’. Lindemann experienced sudden fame witch lead to several
awards and a distinguished professorship at University of Miunich.” (Encyclopedia
Britannica)
Now that there is background on these two mathematicians let’s look at the man who made at
least Wantzel’s proof possible by providing the foundation of a theory in finite fields called
Galois Theory. “Evariste Galois is a French mathematician who was born in Bourg-la-Reine,
France. He was educated at home until 1823 when he entered the College Royal Louis-le-Grand
Taylor 3
and under the guidance of Louis Richard led him to take up the question of the solution of
algebraic equations.” (Encyclopedia Brittannica).
So back to the three “impossible” constructions, in order to understand why these are
impossible we need to understand what exactly is straight edge and compass construction and
what aspect of Galois Theory are important to the proofs which will be provided in the next
section. First what is straight edge and compass construction according to Euclid? First let’s
define what a compass and straight edge is. “A compass is a tool which can be used for drawing
circles, or arcs thereof, whose radii are sufficiently small and for measuring lengths. A
straightedge is a too which can be used for drawing lines, or segments thereof.” (Warren) Now
using both of these tools there are three simple fundamental tasks as given in Euclid’s The
Elements. “These fundamental tasks are as follows: drawing a line through two given points,
drawing a circle having a given point as its center and passing through another given point, and
setting a plane passing through three given non-collinear points, where one performs tasks based
on the two preceding tasks.” (J. Pahikkala). This is the basic foundation of the compass
construction used by Euclid’s and following these strict rules the three proposed “impossible”
constructions are in fact impossible. Now that the straight edge compass constructions are well
defined, let’s define straight edge and compass construction using theory mainly Galois Theory.
“A ruler and compass construction on a set of points 𝑃𝑜 ⊆ ℝ2 is defined by two operations: (1)
draw a straight line between two points in Po. (2) draw a circle centered at some point p ϵ Po with
radius equal to the distance between two points in Po.” (Davis) Also in order to prove that these
are impossible, the word constructability needs to be defined.
“A point 𝑃𝑜 ⊆ ℝ2 is said to be constructible in one step from Po if p is the intersection of
two circles, two lines, or a line and a circle constructible on Po. If ri = (xi , yi) is
Taylor 4
constructible in one step from Pi-1, the we denote 𝑃𝑖 = 𝑃𝑖−1 ⋃{𝑟𝑖 }. A point rn is said to be
constructible from Po if there exist a finite sequence of points r1,….rn ϵ ℝ2 such that for
all i ϵ {1, …..,n}, ri is constructible in one step from the set 𝑃𝑜 ∪ {𝑟1 , … , 𝑟𝑖−1 }.” (Davis).
Using these definitions we can create point field, which is defined by “Let Po be a set of points in
ℝ2 . Then the point field of Po, denoted Ko, is the smallest subfield of ℝ containing every
coordinate of every point in Po. If some point ri = (xi , yi) is constructible in one step from Pi-1,
then the point field Ki of 𝑃𝑖 = 𝑃𝑖−1 ⋃{𝑟𝑖 } is the smallest subfield of ℝ containing Ki-1, xi, and yi.
Using these definitions the following key theorem results:
“If x is a number obtainable from the rationals using only addition, subtraction,
multiplication, division, and the taking of square roots, then x is a solution to some
polynomial equation with rational coefficients. Moreover, if one factors out irrelevant
factors from this equation until one gets down to an "irreducible" polynomial equation
(one that can't be factored any further and still have rational coefficients), the degree of
this polynomial will always be a power of 2.” ("University of Toronto ")
Using this theorem, one can prove as did Wantzel and Lindemann that doubling a cube, squaring
a circle, and trisecting an angle is impossible.
So below are two different ways one can prove that doubling a cube is “impossible”. The
first comes with a simple explanation of roots. “Doubling a cube is impossible because if you
start with a cube of side length 1, you would need to construct a cube whose side length is the
cubed root of 2. But the cubed root of 2 is a solution to the irreducible equation 𝑥 3 − 2 =
0 whose degree, 3, is not a power of 2.” ("University of Toronto "). The other proof has a little
more sophiciation which includes a theorem which concludes from the above definitions which
Taylor 5
is as follows: “Let 𝑃𝑜 ⊆ ℝ2 , and let Ko be it’s point field. Then for all constructible points r =
(x,y), the degrees [Ko(x): Ko] and [Ko(y): Ko] are powers of 2.” (Davis) Now using this theorm
and the above definition’s:
“Given a cube of volume V, a cube of volume 2V is impossible to construct using rulers
and compasses. Proof: If we have a cube, the we have a side of the cube, and we may,
with no loss of generality, assume that one side of the cube is the line between points (0,
0) and (1,0). The volume of such a cube is 1. So constructing a cube of volume 2 would
be equivalent to constructing some point (α,0) such that α3=2. However, the smallest field
containing 0 and 1 is ℚ, and the minimum polynomial of α over ℚ is α3-2. This
polynomial has degree 3, so we have [ℚ(𝛼): ℚ] = 3. But by the above theorem, if (α,0)
is constructible from {(0,0),(1,0)}, then [ℚ(𝛼): ℚ] must be a power of 2. This is
contradictory; therefore, such a point (α,0) cannot be constructed, and we cannot
construct a cube of volume 2.” (Davis)
Now the second “impossible” construction using straightedge and compass construction and
field theory is the the squaring of a circle. Again, provided are two proofs of this construction.
The first proof: “Squaring a circle is impossible because if you start with a circle of radius 1 you
would need to construct a square whose side lengh is √𝜋. But this is a so-called “transcendental
number”: it is not the solution to any polynomial equation with rational coefficients, let alone
one whose degree is a power of 2.” ("University of Toronto ")The second proof:
“Given a circle of area A, a square of area A is impossible to construct using rulers and
compasses. ‘In the proof of this, we will use without proof the fact that π and √𝜋 are
transcendental over ℚ.’ Proof. We can, with no less of generality, assume that our circle
Taylor 6
is the unit circle centered at (0,0). The area of this circle is π, and construction a square
with are π is equivalent to constructing a point (√𝜋, 0). The smallest field containing 0
and 1 is ℚ, so the point field obtained from adjoining (√𝜋, 0) to ℚ is ℚ(√𝜋). However
by the theorem, [ℚ(√𝜋): ℚ] must be a power of 2, and √𝜋 must be algebraic over ℚ,
which is clearly not true. Therefore such a point (√𝜋, 0) cannot be constructed; hence we
cannot construct a square of area π.”(Davis)
Third and final “impossible” construction is the one of trisecting an angle of arbitrary degree.
This proof requires a proof by example.
“Suppose you could construct a 20∘ angle, then the number cos(20∘)would be
constructible (you can just drop a perpendicular from a point on a line at 20∘ to the
horizontal, distance 1 from the origin). However, you can show that α=cos(20∘) is a root
of the equation 8x3−6x−1=0 (by expanding cos(60∘) in terms of cos(20∘) using the
addition formula). It is easy to show that this has no rational roots, and so the roots are
not constructible. This means that we couldn't have constructed a 20∘ angle, because then
we would be able to construct cos(20∘) which is impossible. So a 60∘ angle cannot be
trisected.” (Goodman)
This is the only proof which will be provided in this paper of the trisection of an angle. So
provided are the proofs of impossibility in construction using a straightedge and compass.
In conclusion, field theory is used a lot in the world of geometry and the construction of
various shapes and objects. Polynomial roots are key to this theory. During the past couple of
century’s these questions have bugged mathematicians and a deeper understanding of field
theory with Galois’s findings have provided a way to prove that at least three things cannot be
Taylor 7
constructed using only a straight edge and compass. These three “impossible” constructions are
as proven the doubling of a cube, the squaring of a circle, and the trisecting of an angle.
Taylor 8
Work Cited
Davis, Isaac. "Understanding Ruler and Compass Constructions with Field Theory." . University of Chicago, n.d.
Web. 22 Apr 2012. <www.math.uchicago.edu>.
"Évariste Galois." Encyclopædia Britannica. Encyclopædia Britannica Online Academic Edition. Encyclopædia
Britannica Inc., 2012. Web. 18 Apr. 2012.
<http://www.britannica.com/EBchecked/topic/224587/Evariste-Galois>.
"Ferdinand von Lindemann." Encyclopædia Britannica. Encyclopædia Britannica Online Academic Edition.
Encyclopædia Britannica Inc., 2012. Web. 18 Apr. 2012.
<http://www.britannica.com/EBchecked/topic/341841/Ferdinand-von-Lindemann>.
Freeman, Larry. "Pierre Laurent Wantzel." Fermat's Last Theorem. N.p., 25 10 2008. Web. 22 Apr. 2012.
<fermatslasttheorem.blogspot.com>.
Goodman, Dan. "An Introduction to Galois Theory." nrich: enriching mathematics. University of Cambridge, 2002.
Web. 22 Apr 2012. <nrich.maths.org>.
"impossible." Dictionary.com Unabridged. Random House, Inc. 22 Apr. 2012.
<Dictionary.com http://dictionary.reference.com/browse/impossible>.
J. Pahikkala, Warren Buck, Raymond Puzio, Roger Lipsett. "geometric constructions by Euclid" (version
17). PlanetMath.org.
Suzuki, Jeff. "A Brief History of Impossibility." Mathematics Magazine. 81.1 (2008): 27-28. Web. 22 Apr. 2012.
<mathdl.maa.org>.
"The Three Classical Impossible Constructions of Geometry." University of Toronto . N.p., 04 19 1999. Web. 22
Apr 2012. <www.math.toronto.edu>.
Warren Buck. "compass and straightedge construction" (version 19). PlanetMath.org.
Taylor 9