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Constructible Numbers and Galois Theory Zainab Ahmed

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Constructible Numbers and Galois Theory Zainab Ahmed Constructible Numbers and Galois Theory Zainab Ahmed Important Definitions Algebraic Numbers All rational numbers are constructible numbers, which are all algebraic numbers. Algebraic numbers can be complex orp real. A complex algebraic number, for example, can be i. An example of a real algebraic number is 2. The root of this polynomial equation is algebraic: n n bnx + bn − 1x − 1 + ::: + b1x + b0 = 0 where bi are algebraic numbers. Constructible Numbers A constructible number is a number represented by finite additions, subtractions, multiplications, divisions, and square roots of integers. These numbers make up line segments that can only be formed with a straightedge or compass. p An example of a constructible number is 2. First off, groups, rings, and fields need to be defined. They can be revealed through Galois' Associative and Commutative Property Theorem. Groups A group is a finite set G equipped to a; bG the product a × b and satisfies i/ (a × b) × c = a × (b × c) for any a; b; c G ii/ There exists e in G such that a × e = e × a = a 8aG (one shows that e is unique) iii/ For any a in G, there exists a0 G : a × a0 − a0 × a = e, where a0 is a's reciprocal. Examples: (R; +); (R∗; x); (C; +); (C∗; x); (Z; +); (Q; +); (Q∗; x) where R are rational numbers, C are complex numbers, Z are integer numbers, and Q are rational numbers. the ∗ symbol indicates that the group are of nonzero numbers. Also if the following additional property holds: iv/ a × b = b × a 8a; bG then the group is called abelian or commutative. Rings A ring is a set R with two operations: +, x such : i/ (R; +) is an abelian group ii/ (ab)c = a(bc) 8a.b.cR iii/ a(b + c) = ab + ac 8a.b.cR If we also require that: iv/ 91R : a × 1 = 1 × a = a 8aR v/ ab = ba8a; bR then we have a commutative ring with identity. QED Fields Finally if 8aR9a0 : aa0 = 1 then R is called a field. A field can contain rational, complex, real, and algebraic numbers. Within a field, zero does not have a reciprocal. A field is a division ring with commutative multiplication. Examples: Q; R; Zp, where p is always prime. Extension Fields An extension field are when a field F is a subfield of K which is denoted as K=F . For example, complex numbers is an extension field of real numbers which is an extension field of rational numbers. An extension field degree is something that indicates the measure of the size of a filed extension. If K=F is the extension field, then K is considered as a vector space, or the product or sum of scalar vectors, over F, the field of scalars. An extension field degree is noted as: [K : F ] = dimF K A simple extension is when only one new element from the yield of the rationals of a complex number is added. To sum up the essence of a simple extension, it is written in the following way: f(a) F (a) = g(a) where a is an element within the larger filed F , f; g are polynomials within the field F , and where g(a) 6= 0 2 F 0 The following formula shows that for some power of n, the nth power of an can be written as a linear combination. The powers of a is always less than n. a is considered to be an algebraic number within F and F (a) is the algebraic extension. The smallest integer n that satisfies the summation above is the extension field degree. The summation 2 of p(x) = xn is called an extension field minimal polynomial. An extension field minimal polynomial is where p(a) = 0. Extensions can then turn into transcendental extensions when there is no such integer of n where p(x) = xn. Background Information and History The Galois theory is named after Evariste Galois. He lived from October 25, 1811 to May 31, 1832. His father was a small town mayor. His mother was a lady who taught Galois politics and how to be a noble. Even though he was interested in politics, he went to school to study mathematics. Being rejected form Ecole Polytechnique, the most prestigious institution for mathematics in France at the time, he went to Ecole Normale, a poor university for mathematics at the time. He was a French mathematician who was able to understand how to compute a polynomial with radicals in his teens. He is known for his works on the theory of equations and Abelian integrals. His work led up to the two major branches of abstract algebra, Galois theory and group theory. He also introduced a infinite field, also known as Galois field. This field is usually shown as fields containing integers modp where p must always be a prime number. Galois Theory, the most notable of mathematics founded by Evariste Galois, shows the algebraic solutions to to a polynomial equation. Galois found that the solutions to polynomial equations are associated with the permutations, or the sequence or order of the set, of the roots of the polynomial equations. This discovery a started with the question of finding the formula for the roots of a fifth or higher degree polynomial equation with only the use of algebraic computations (addition, subtraction, multiplication, and division) and radicals. It also answers why there is a need to define a constructible number, as not everything is geometrically constructible. To go about solving the polynomial equations, Evariste Galois originally applies the theory of equations, which is the analysis of the algebraic solutions to polynomial equations. He found that radicals can represent the solutions to a polynomial equation if the series of commutative subgroups of a Galois group, or the permutations, is found. Involving Galois' last days and death, a myth states that at one dinner, Galois disrespected the king and was sentenced to prison. In prison, he constructed a proof for the solutions of the roots of the fifth degree polynomial. When he was released from prison, Galois was expelled from Ecole Normale. Evariste Galois first submitting his memoir on the theory to the Paris Academy of Sciences in 1830 at the age of 18. However, due to his lack in written explanation of all details of the theory, his memoir was thought to be too sketchy and unrealistic at the time. Therefore, this memoir was not published. He then romanced a woman, but was then challenged to a duel over her hand. The night before the duel, he wrote down all of his mathematical thoughts to be passed on and sent them to his friend, as though he knew that he would die in the upcoming duel. However, after Galois died, the paper from the night before his duel was lost. Later the document was found; yet, he is not credited to be the first one who found the proof for the solutions of the roots of the fifth degree polynomial. 3 Evariste Galois's memoir of the theory of solvability by radicals was not published until 1846 by Joseph Liouville, a prestigious mathematician at the time. Joseph added onto Galois' theory and clarified his work before publications of the work documents. Yet, many mathematicians had trouble understanding Evariste Galois' work. Application of Constructible Numbers Splitting fields and Multiple Roots Galois Theory Galois Theory simply states: If there exists a one-to-one correspondence between two subgroups and subfields such that G(E(G0)) = G0 where G0 is a larger field than G E(G(E0)) = E0 where E0 is a larger field than E then E is said to have a Galois theory. Galois Theory can be done though permutations on quadratic equations only where the coefficients are rational numbers. The Galois group of quadratic equations have two kinds of permutations: the identity permutation and the transposition permutation. The identity permutation does not effect the coefficients of the equation at hand because it notes every element of the set itself. The transposition permutation is where two elements within the set are switched, changing the coefficients and essentially the solutions to the quadratic equations. 4.
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