By Falko Lorenz

From Math stories: "This is a captivating textbook, introducing the reader to the classical elements of algebra. The exposition is admirably transparent and lucidly written with purely minimum must haves from linear algebra. the recent thoughts are, at the least within the first a part of the publication, outlined within the framework of the improvement of conscientiously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on buildings with ruler and compass of their algebraic surroundings within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The e-book ends with an appendix containing routines and notes at the earlier elements of the ebook. despite the fact that, short old reviews and recommendations for additional analyzing also are scattered during the text."

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**Example text**

Iii) Every nonconstant polynomial in C ŒX is completely decomposable into linear factors. (iv) If E=C is an algebraic ﬁeld extension, E D C . Proof. (i) ) (ii): Let f 2 C ŒX be irreducible. ˛/ D 0. X ˛/, necessarily with 2 C . (ii) ) (iii): By (ii), only linear polynomials can appear in the prime factorization of f 2 C ŒX . (iii) ) (iv): Let E=C be algebraic. ˛/ of any ˛ 2 E is irreducible, hence linear, by (iii). Therefore ˛ 2 C . The algebraic closure of a ﬁeld 57 (iv) ) (i): Suppose f 2 C ŒX is nonconstant.

From now on we assume R is a unique factorization domain. We denote by K D Frac R the fraction ﬁeld of R. Now let be a given nonzero prime of R. The corresponding -adic valuation w W K ! [ ޚf1g in R (page 40) can be extended to a map w W KŒX ! f / is the exponent of the highest that ﬁts in all coefﬁcients of f . f / for c 2 K; f 2 KŒX : The springboard for the proof of Gauss’s Theorem is provided by the next result: F5. Let R be a unique factorization domain and ¤ 0 a prime in R. h/: Proof.

1 ˛ C ˇ/ D K. 2 ˛ C ˇ/ DW L: Then . 1 ˛ C ˇ/ . 2 ˛ C ˇ/ D . 1 2 /˛ lies in L, and therefore so does ˛, and likewise ˇ. It follows that E D L D K. 1 ˛ C ˇ/, so that E=K is simple (with D 1 ˛ C ˇ as a primitive element). ˛; ˛n /. For K a ﬁnite ﬁeld the assertion follows from the fundamental theorem of the theory of ﬁnite ﬁelds, which we will study later (Theorem 2 in Chapter 9). ˜ 4 Fundamentals of Divisibility Throughout this chapter, R stands for a commutative ring with unity. Much of the content of this chapter is probably familiar to you from earlier courses.