A new approach to linear filtering and prediction problems by Kallenrode

By Kallenrode

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The field ‫ ރ‬of complex number is algebraic closed; this is proved through analysis, function theory or algebra (see Volume II for the latter). F2. The following statements about a field C are equivalent: (i) C is algebraically closed. (ii) Every irreducible polynomial in C ŒX  is linear (that is, of degree 1). (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.

We single out a special case: 42 4 Fundamentals of Divisibility F15. If E; F are fields and ' W E ! F is a homomorphism of rings with unity (meaning that '1E D 1F ), then ' is injective, and so provides an isomorphism between E and a subfield E 0 of F . Definition 10. Let R be a commutative ring with unity. Two ideals I1 ; I2 of R are relatively prime if I1 C I2 D R; in other words, when there exists a 2 I1 and b 2 I2 such that a C b D 1. The product I1 I2 of two ideals I1 ; I2 of R is the ideal of R generated by all products xy, where x 2 I1 and y 2 I2 ; thus is consists of all finite sums of such products.

Lemma. (a) For I1 ; I2 relatively prime ideals of R we have I1 I2 D I1 \ I2 . (b) If an ideal I1 of R is relatively prime to each of the ideals I2 ; I3 ; : : : ; In of R, it is also relatively prime to the product I2 I3 : : : In . Proof. (a) From 1 D a C b with a 2 I1 and b 2 I2 we conclude by multiplying with an arbitrary c 2 I1 \ I2 that c D ca C cb 2 I1 I2 . (b) By assumption there exists for each i D 2; 3; : : : ; n an element ai 2 I1 and a bi 2 Ii such that 1 D ai C bi . ai C bi / 2 I1 C I2 I3 : : : In : ˜ 1D i F16 (Chinese Remainder Theorem).

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