Galois Theory and Its Applications

This work laid the foundation for Galois Theory, a major branch of algebra. Galois was bored with education at age 14, so he left school and took an interest in mathematics. In 1929, he published a paper on continued fractions and made discoveries on the Theory of Polynomial Equations that were reviewed by Augustin Louis Cauchy, the developer of the Cauchy Criterion. In 1830, Cauchy submitted these discoveries to Joseph Fourier, of Fourier series fame, for consideration for the Academy’s Grand Pix Award. Fourier died and Galois discoveries were lost. During his imprisonment, he kept on developing his mathematics. While still in prison, Simeon Poisson, who developed Poisson distribution, viewed his work on the theory of equations but he declined to publish it.

Galois was disappointed but kept on improving his ideas in prison until his release in April, 1932. After he was released, he decided to start publishing his work privately through his friend Auguste Chevalier. Later that year, he was involved in a conflict that led to him being shot in the abdomen and died. • Studied elliptic functions and considered the integrals of most general algebraic differential, called the abelian integrals. • He realized that the algebraic solution for a polynomial is related to a group of permutations called the Galois group, which relates to the roots of the given polynomial. He also found out that this equation can be solved by radicals if you can find the series of subgroups. This can be applied only to members of the Galois group.

GROUPS A group (G,*) is a set G which is closed under a binary operation * such that: • For all a, b, c ϵ R, we have (a * b) * c = a* (b * c) associate of * • There is an element e in G such that for all x ϵ G, e * x = x * e = e identity element e for *. The map ψ is called the group operation. The element e is unique. For, if e1 ∈ G such that e1x = xe1 = x, ∀ x ∈ G, we have, in particular, e = ee1 = e1. The element e is called the identity element of G. The element x’ is unique such that if x’’ ∈ G, then x’’x = xx’’ = e. A homomorphism f: G → G’ is called an isomorphism, if there exists a homomorphism g: G’ → G such that g * f = IG and f *g = IG(0).

We then write G ≈ G0. An isomorphism f: G → G is called an automorphism. A homomorphism is an isomorphism iff it is one-one and onto. SUBGROUPS AND QUOTIENT GROUPS A subgroup H of a group G is a non-empty subset H of G such that if x, y ∈ H, then x−1y ∈ H. SYMMETRIC GROUPS AND SOLVABILITY Let Sn be the symmetric group of degree n. An element of order r is a permutation σ such that there exist r distinct integers x1,. xr (1 ≤ xi ≤ n) for which σ(x1) = x2,. σ(xr−1) = xr, σ(xr) = x1 and σ(x) = x for x =6 xi, 1 ≤ i ≤ r. We then write σ = (x1,x2,. RINGS AND HOMOMORPHISMS A commutative ring A is said to be a field if A∗ = A−{0} is a group under multiplication, so that every non-zero element is a unit.

A subring R of a field K is called a subfield of K if the ring R is a field. Any intersection of subfields of K is again a subfield. If S is a subset of K, then the intersection of all subfields of K containing S is called the subfield generated by S. For any ring A, the identity map is a homomorphism. bn,. We make R into a ring by defining the ring operations as follows: f + g = (a0 + b0,. an + bn), fg = (c0,. cn), cn = X aibj, where i+j=n The unit element of R is (1,0,0,. The mapping f: A → R defined by f(a) = (a,0,. Let B be a ring and A a subring of B.

For any α B we define a map ψ: A[X] → B by setting ψ(PaiXi) = Paiαi. It is easily verified that ψ is a ring homomorphism. Therefore ψ(A[X]) = A[α] and if f(X) = PaiXi ∈ A[X], then ψ(f) = f(α)(= Paiαi). Then we can deduce that α is a root of f if f(α) = 0. S is a base of V if and only if every element v ∈ V can be uniquely written as finite sum v = Pλisi, λi ∈ K, si ∈ S. ALGEBRAIC EXTENSION Let K be a field, a set with two binary operations, and k a subfield of K. Then k will be called an extension of K and written K/k. Two extensions K/k, K0/k are said to be k-isomorphic if there exists an isomorphism σ of K onto K0 such that σ|k is the identity.

σ is then called a k-isomorphism. Case (i) – If Ker φ = (0), i. e. α is not a root of any non-zero polynomial over k, we say that α is transcendental over k. Then, the one-one homomorphism φ:k[X] → k(α) can be extended to a one-one homomorphism of k(X), the quotient field of k[X], onto a subfield of k(α). The α and k, in the subfield, should coincide with k(α). We note that if α ∈ K is algebraic over k it is algebraic over any field L such that K ⊃ L ⊃ k. SEPARABLE EXTENSIONS Let k be a field. An irreducible polynomial f ∈ k[X] is called separable if all its roots are simple, otherwise, f is called inseparable. A non-constant polynomial f ∈ k[X] is called separable if all its irreducible factors are separable.

Let K/k be an algebraic extension. If σi be an extension of σi to an automorphism of N; we assume that σi *τj are distinct. Suppose σi * τj(x) = σi * τs(x) for every x ∈ L. Therefore, for every x ∈ K, there exist σi(x) = σr(x), which implies that i = r. Therefore, τj(x) = τs(x) for every x ∈ L implying that j = s. Let θ be any k-isomorphisms of L into N, Clearly θ|K = σi for some i. Thus any element of F satisfies the polynomial Xpn − X ∈ Z/(p)[X]. Since F has npn elements, we can conclude that F is the splitting field of the polynomial Xp −X over Z/(p). Since all the roots of this polynomial are distinct, then F is a separable extension of Z/(p).

FUNDAMENTAL THEOREM OF GALOIS THEORY In most basic forms, this theorem assumes that given a finite field extension E/F, there must be a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. Intermediate fields K must satisfy F ⊆ K ⊆ E. APPLICATION OF GALOIS THEORY 1. Galois, while a teen, determined the necessary and sufficient conditions for solving quintic polynomials, degree five or higher, by radicals. This was done by factorizing the quantic polynomial to lower degrees by use of group theory. Solving the quantic polynomials was a great achievement because it is a very broad area in number theory. Number theory has many uses in modern world mostly in encryption of secure and confidential documents. e. debugging. Galois Theory is applied in the encryption of cyphers used in cryptography.

In cryptography, the difficulty of the logarithm problem in finite fields forms the basis of several widely used protocols. A good example is the Diffie-Hellman protocol. B. A First Course in Abstract Algebra, Addison Wesley, Boston. Rotman, J. Galois Theory, Springer, New York. Stewart, I.