I have a good applied math background, including linear algebra and gaussian elimination. First part I don't totally understand is: Prasad’s work is for solving linear systems over finite fields, specifically F_p, with p prime.
Re-reading your question, since you are already familiar with linear algebra, the thing you need to know about is finite fields. But read the whole thing, it's in a logical progression.
OK ... so a linear equation is one of the form:
ax + by + ... + dz = n
where the a, b, d and n are constants and the x, y, z, etc are the variables. You may be familiar with these when the constants are real numbers, but they can be from any field (more about that in a minute).
An example of a linear equation is
3x + 5y - 2z = 1
So a system of linear equations is when you have lots of these. In large systems you may have literally millions of variables.
So what's this about a "field"?
Well, in general the constants don't need to be real numbers. There are properties that we want, such as addition, multiplication, subtraction and division, but there are systems that have these sorts of things that are not the real numbers. These are called fields.
For example, the rational numbers have all the properties we might want. We can add, subtract, divide and multiply rationals, and things work the way we would want.
But another example is the integers modulo 5. The numbers 0, 1, 2, 3 and 4 have all the properties we want. We can divide by 2, for example, but multiplying by 3, because 2x3=6, which is 1 (modulo 5).
So we can have a field (a system with all the operations we want) that only has a finite number of elements. There are others, but the most common ones are to take the integers modulo some prime number. The proof that we get a field is not har, and is actually interesting, but some people find it tedious.
So Prasad's work is for solving systems of linear equations where the constants are not real numbers, but come from a finite field.
Finally, one of the most common finite fields in the area of computing is the field that has just two elements - 0 and 1.
The other answers in this sub-thread are pretty good. Again, if you want to understand you will need to "read like math", but once you've done so, come and ask again if you have any questions.
You've probably used Gauss-Jordan elimination over the reals. What properties of real numbers make Gauss-Jordan elimination work?
The real numbers have addition and multiplication operations which each obey commutative, associative, identity, and inverse laws; and multiplication is distributes over addition. These properties are all that's required for Gauss-Jordan elimination to work.
(We don't need other properties of real numbers, such as properties involving convergence of sequences, which you may have studied in calculus or real analysis class.)
A set and binary operators + and * which have all of these properties, are called a "field" [1]. The real numbers, rational numbers, and complex numbers are probably your most familiar examples of fields.
When talking about a field like "the field of rational numbers, Q", you're actually referring to a data structure (Q, +, *) which includes definitions of the addition and multiplication operators.
Since most fields have "standard" addition and multiplication operators, often mathematicians drop them from discussion. Kind of like, when you're discussing a program with a colleague, you say, "Add the offset to the base," your colleague doesn't ask, "Add them as integers or floating-point numbers? What should I do if they overflow?" The answers to these questions are almost always clear from context, and also almost always near-universally standardized, at least within a particular programming language.
Anyway, as other posters have noted, the integers modulo p, where p is prime, are actually a field. It's obvious that Z_p has most of the properties of a field, but multiplicative inverse is interesting. The proof of existence of multiplicative inverses mod p -- and a fast O(log(p)) implementation of finding them -- is the Euclidean algorithm, one of the oldest algorithms known to mankind. [2]
Do you know all the words in that sentence, and just can't seem how they work together to create meaning, or do you have problems with any specific word?
