By Joseph Muscat

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**Example text**

Let m/ be the minimum polynomial of a/ over K and let / = mi—ms. Each m/ is irreducible over K and has a zero otj e L so by normality each ro/ splits over L. Therefore / splits over L. Since L is generated by K and the zeros of / , it is a splitting field for / over KM The additional concept of separability is required for fields of non-zero characteristic. An irreducible polynomial / over afieldK is separable over K if it has no multiple zeros in a splitting field. {x - a n ) where the a/ are all different.

If dg = 0 or 1 then g is automatically irreducible. If g > 1, either g is irreducible or g = hj where dh and dj < dg. By induction h andy are products of irreducible polynomials; therefore g is also such a product. 1-10: For any field K, the factorization of polynomials over K into irreducible polynomials is unique up to constant factors and the order in which the factors are written. ygs are irreducible polynomials over K. If all the// are constant then/e K so that all of the gj are constant. Otherwise we may assume that noft is constant by dividing out all the constant terms.

Any simple algebraic extension is thus finite. However, the converse is not true. In this connection an extension L:K is algebraic if every element of L is algebraic over K. ,^). 2 Galois Theory: Solubility of Algebraic Equations by Radicals Group theory was invented by Galois to study the permutations of the zeros of polynomials. Thus any polynomial f(x) has a group of permutations of its zeros, now called its Galois group, whose structure is closely related to the methods required for solving the corresponding polynomial equation f(x) = 0.