1. Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.

2. Expressed algebraically, that same equation would be:

3. A finite field F is not algebraically closed.

4. A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.

5. Quasi-algebraically closed fields are also called C1.

6. The fact that the complex numbers are algebraically closed is required here.

7. Now lets think about how we can represent that algebraically

8. Expressed algebraically, $g(u) equals: SL = (TT)/(cosine($g(u)).

9. Using invariant theory, we obtain a smaller system of algebraically independent equations.

10. There is some ambiguity here if K is not algebraically closed.

11. A field with no nontrivial algebraic extensions is called algebraically closed.

12. Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.

13. Synonyms for Computationally include mathematically, algebraically, arithmetically, numerically, statistically, geometrically, trigonometrically, topologically

14. So we just have to algebraically manipulate these equations into this form.

15. Synonyms for Arithmetically include mathematically, algebraically, numerically, statistically, geometrically, trigonometrically, topologically, algorithmically

16. The post-Newtonian expansion is analyzed for an algebraically extended theory of gravity equivalent to a theory with a real, nonsymmetric metric, previously referred to as Algebraically Extended Hilbert Gravity (AHG).

17. The extensive matrix operations are performed algebraically on the digital computer using index coding.

18. More generally, no polygon whose vertex coordinates are algebraically independent has an equidissection.

19. The full theorem generalizes to any algebraic variety over an algebraically closed field.

20. We took this kind of crazy looking thing, and all we did is algebraically manipulate it.

21. Singular values are similar in that they can be described algebraically or from variational principles.

22. So just algebraically manipulating it, we were able to put it into slope- intercept form.

23. The Zariski topology on an algebraic variety (over an algebraically closed field) is T1.

24. Said values (U14, U15) are algebraically added in an adder (51) during each of the bit intervals.

25. And if you think about it algebraically you might get rid of the Y here.

26. Alternatively, images obtained using at least two wavelengths can be algebraically combined for noise reduction.

27. The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.

28. The general Gauss-quadratures may be determinated algebraically if sufficiently many fix-elements of the interpolating operator are known.

29. The strain energy at each particle can be expressed algebraically in terms of the extension ratios and other measurable quantities.

30. Now, we can literally just algebraically manipulate this guy right here to put it into our slope intercept form.

31. The usual concepts of the relational model can thus be presented in a uniform way and interpreted algebraically.

32. It is argued that 2J decreases algebraically as the van der Waals solute–solvent interactions increase in magnitude.

33. When this occurs, the subtraction of the wet-bulb temperature from the dry-bulb temperature shall be done algebraically.

34. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

35. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.

36. From the N dependence of the three sublattice or helical magnetization, the spin correlation is conjectured to decay algebraically as .

37. It is shown that such symbolic algebraic thinking includes finding an algebraic approach, acting intentionally in an algebraic way, and reflecting algebraically.

38. In this paper it will be shown, in which ways algebraic symbols enable learners to think algebraically in non-routine problems.

39. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic.

40. In this paper this transformation is reduced to the multiplication of polynomials, which can easily be performed algebraically on a computer.

41. The mechanics of combined flow and fracture are expressed algebraically, and in terms of work areas both under load and unloaded.

42. In the first section of this paper, we consider the case of binary cubic forms and the base fieldK is algebraically closed.

43. However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent.

44. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers.

45. The formulae expressing algebraically a computing programme may contain brackets (Chap 5) or polynomials in several variables put in the normal form (Chap.

46. These equations have been given a block-diagram interpretation, and the block diagram has been transformed algebraically to assume rhe standard form of a regulator.

47. The electrical circuit analyzer linearly and algebraically solves an equation system including the admittance matrix for analyzing at least apart of the electrical circuit.

48. To put it algebraically, 2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2.

49. And it actually turns out it's a sum of things we already know, and we just have to manipulate this a little bit algebraically.

50. Formulae describing some aspects of our study algebraically are presented in Appendix B. We also indicate what cautions are needed in interpreting our results.

51. Spherical equivalent is determined by algebraically adding half the cylinder part of the correction to the spherical part of the correction, (cylinder/2) + sphere.

52. So once again, we just have to algebraically manipulate it so that the x's and the y's are both on this side of the equation.

53. In the check equations, each side of the equal sign is algebraically equivalent but cannot be verified exactly since the readings mi are obtained experimentally.

54. A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental.

55. This is a special case of Schanuel's conjecture, but so far it remains to be proved that there even exist two algebraic numbers whose logarithms are algebraically independent.

56. He, with Harry P. Allen, used Hopf algebras to prove in 1969 a famous 25-year-old conjecture of Jacobson about the forms of generalized Witt algebras over algebraically closed fields of finite characteristic.

57. The quantities on each side of the equal sign (=) of the check equations are obtained from combinations of the linear equations of condition that are equivalent to one another algebraically.

58. The driver unit includes a shift amplifier (22) for algebraically superimposing a pulsed analog data voltage ADV (t) corresponding to incoming digital data pulses, and the shift voltage SV (t).

59. ‘The problem comes when you try to think algebraically, or Arithmetically, about the Gauss problem.’ ‘But the resources necessary to sustain that increase could be multiplied only Arithmetically, adding only a fixed amount every year.’

60. ‘The problem comes when you try to think algebraically, or Arithmetically, about the Gauss problem.’ ‘But the resources necessary to sustain that increase could be multiplied only Arithmetically, adding only a fixed amount every year.’

61. Algebraically, this scenario has the form Pseudo-scenario C = PST2010C â€" 2*(PST2010A â€" PST2010B) Given these adjustments the only difference between these two new “pseudo-scenarios†will be the spatial distribution of the SO2 emission reductions in Ontario and Quebec.

62. In fact, when C C and F F are well-behaved, then Algebras over an endofunctor F F are equivalent to Algebras over a certain monad, the algebraically-free monad generated by F F (Pirog, Gambino-Hyland 04

63. There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π).

64. So all I did is algebraically manipulated this top equation up here, and when I did that, when I solved essentially for y, I got this right over here, which is the exact same thing as the second equation.

65. More algebraically, one can abstract the notion of a Poincaré complex, which is an algebraic object that behaves like the singular chain complex of a manifold, notably satisfying Poincaré duality on its homology groups, with respect to a distinguished element (corresponding to the fundamental class).

66. I just algebraically manipulated it has a y intercept at - 2 and has a slope of 1/ 2 so that means that for every one it goes over, it only goes a half up or for every 2 it goes over it only goes 1 up.

67. Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is an abelian variety – that is, it has a multiplication defined algebraically, with respect to which it is an abelian group – and O serves as the identity element.

68. The device further comprises: a further reader (40), arranged to read second magnetic signals belonging to the electromagnetic noise, an adder component (25) arranged to algebraically subtract the amplified second magnetic signals from the amplified first magnetic signals, and a converter (16) arranged to convert the resulting signal into a digital signal representing the read magnetic patterns.